Search: keyword:new
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A361776
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Expansion of g.f. A(x) satisfying x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * A(x)^n * (A(x)^n + x^n)^n.
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+0
0
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1, 1, 6, 33, 198, 1204, 7522, 48270, 316281, 2110018, 14293494, 98054885, 679735489, 4753912524, 33504984427, 237767467381, 1697719206178, 12188097989345, 87913304459342, 636736565338008, 4628839922257617, 33767007201285762, 247145222148251103, 1814452818239003585
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OFFSET
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0,3
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COMMENTS
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First negative term is a(51) = -47152346702575235627205086026135269902810693.
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * (x*A(x))^n * (A(x)^n + x^n)^n.
(2) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * (x*A(x))^(n*(n-1)) / (A(x)^n + x^n)^n.
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EXAMPLE
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G.f.: A(x) = 1 + x + 6*x^2 + 33*x^3 + 198*x^4 + 1204*x^5 + 7522*x^6 + 48270*x^7 + 316281*x^8 + 2110018*x^9 + 14293494*x^10 + ...
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PROG
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(PAR) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(x*Ser(A) - sum(m=-#A, #A, (-1)^m * x^m * Ser(A)^m * (Ser(A)^m + x^m)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign,new
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AUTHOR
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STATUS
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approved
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A362219
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Decimal expansion of smallest positive solution to tan(x) = arctan(x).
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+0
0
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4, 0, 6, 7, 5, 8, 8, 8, 6, 5, 7, 6, 5, 8, 6, 2, 7, 9, 0, 9, 1, 7, 0, 8, 5, 0, 2, 5, 3, 1, 2, 4, 1, 1, 3, 1, 9, 0, 6, 8, 3, 0, 0, 6, 7, 4, 4, 9, 3, 9, 5, 7, 9, 2, 2, 6, 3, 7, 2, 6, 3, 4, 3, 6, 5, 5, 1, 4, 6, 5, 8, 6, 2, 6, 6, 0, 5, 4, 7, 1, 0, 1, 5, 5, 9, 0, 2, 8, 2, 3, 7, 7, 0, 4, 4, 0, 0, 1, 1, 6, 8, 2, 0
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OFFSET
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1,1
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LINKS
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FORMULA
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EXAMPLE
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4.067588865765862790917085025312411319068300674493957922637263436551...
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MATHEMATICA
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RealDigits[FindRoot[Tan[x] == ArcTan[x], {x, 4}, WorkingPrecision -> 105][[1, 2]]][[1]]
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PROG
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(PARI) solve(x=4, 4.5, tan(x)-atan(x)) \\ Michel Marcus, Apr 12 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A362220
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Decimal expansion of smallest positive root of x = tan(tan(x)).
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0
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1, 3, 2, 9, 7, 3, 1, 2, 2, 0, 6, 7, 8, 9, 4, 5, 5, 1, 5, 7, 3, 7, 1, 4, 6, 0, 6, 5, 5, 8, 4, 6, 4, 8, 5, 8, 9, 6, 0, 4, 8, 2, 9, 8, 5, 7, 4, 9, 0, 3, 8, 0, 4, 3, 6, 7, 5, 1, 2, 4, 6, 4, 5, 7, 9, 7, 9, 9, 7, 8, 0, 4, 7, 0, 6, 0, 1, 4, 3, 2, 0, 4, 5, 8, 3, 8, 2, 3, 7, 1, 3, 6, 9, 5, 1, 6, 2, 4, 8, 8, 4, 3, 6
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OFFSET
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1,2
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LINKS
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FORMULA
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EXAMPLE
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1.329731220678945515737146065584648589604829857490380436751246457979...
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MATHEMATICA
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RealDigits[FindRoot[Tan[Tan[x]] == x, {x, 1.3}, WorkingPrecision -> 105][[1, 2]]][[1]]
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PROG
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(PARI) solve(x=1.32, 1.35, tan(tan(x)) - x) \\ Michel Marcus, Apr 12 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A361358
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Expansion of x*(2 - x)/(1 - 5*x + 3*x^2 - x^3).
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3
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2, 9, 39, 170, 742, 3239, 14139, 61720, 269422, 1176089, 5133899, 22410650, 97827642, 427040159, 1864128519, 8137349760, 35521403402, 155059096249, 676868620799, 2954687218650, 12897889327102, 56302253600359, 245772287239139, 1072852564721720
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OFFSET
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1,1
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COMMENTS
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This sequence arises in the enumeration of noncrossing caterpillar graphs (A361356). Given a directed edge (A,B) on the spine of the caterpillar where B is not a leaf node, then a(n) is the number of ways to complete the caterpillar using at most n nodes. Nodes cannot be added to A. Equivalently, a(n) is the number of ways to complete the caterpillar using exactly n nodes allowing leaves to be added to the left of A (but not to the right).
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LINKS
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FORMULA
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a(n) = 5*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
G.f. A(x) satisfies A(x) = x*(2 - x + 2*A(x))/(1 - x)^3.
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EXAMPLE
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In the following examples, o is a leaf and 1..n+1 is the spine.
a(1) = 2, a leaf can be added to the left or to the right of the spine:
1---2 1 o
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o 2
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a(2) = a(1) + 7:
1---2 1---2 1---2 1 o 1 3 1 o 1 o
/ | / | \ | | / | | | | /
3---o o---3 o o o---2 2 o 2---3 2---o
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PROG
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(PARI) Vec(x*(2 - x)/(1 - 5*x + 3*x^2 - x^3) + O(x^25))
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CROSSREFS
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KEYWORD
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nonn,easy,new
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AUTHOR
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STATUS
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approved
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A362893
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Number of partitions of [n] whose blocks can be ordered such that the i-th block has at least i elements and no block j > i has an element smaller than the i-th smallest element of block i.
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1, 1, 1, 2, 5, 12, 28, 69, 193, 614, 2103, 7359, 25660, 88914, 309502, 1102146, 4092840, 16046224, 66410789, 286905421, 1273646720, 5729762139, 25881820352, 116872997038, 527375160184, 2384407416357, 10856086444051, 50097994816979, 235937202788389
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OFFSET
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0,4
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LINKS
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EXAMPLE
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a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 1|23.
a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
a(5) = 12: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(6) = 28: 123456, 1234|56, 1235|46, 1236|45, 123|456, 1245|36, 1246|35, 124|356, 1256|34, 125|346, 126|345, 12|3456, 1345|26, 1346|25, 134|256, 1356|24, 135|246, 136|245, 13|2456, 1456|23, 145|236, 146|235, 14|2356, 156|234, 15|2346, 16|2345, 1|23456, 1|23|456.
a(7) = 69: 1234567, 12345|67, 12346|57, 12347|56, 1234|567, 12356|47, 12357|46, 1235|467, 12367|45, 1236|457, 1237|456, 123|4567, 12456|37, 12457|36, 1245|367, 12467|35, 1246|357, 1247|356, 124|3567, 12567|34, 1256|347, 1257|346, 125|3467, 1267|345, 126|3457, 127|3456, 12|34567, 12|34|567, 13456|27, 13457|26, 1345|267, 13467|25, 1346|257, 1347|256, 134|2567, 13567|24, 1356|247, 1357|246, 135|2467, 1367|245, 136|2457, 137|2456, 13|24567, 13|24|567, 14567|23, 1456|237, 1457|236, 145|2367, 1467|235, 146|2357, 147|2356, 14|23567, 14|23|567, 1567|234, 156|2347, 157|2346, 15|23467, 167|2345, 16|23457, 17|23456, 1|234567, 1|234|567, 15|23|467, 1|235|467, 16|23|457, 1|236|457, 17|23|456, 1|237|456, 1|23|4567.
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MAPLE
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b:= proc(n, t) option remember; `if`(n=0 or n=t, 1,
add(b(n-j, t+1)*binomial(n-t, j-t), j=t..n))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..28);
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A362822
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Number of nonisomorphic magmas with n elements satisfying the identities (xy)y = x and (xy)z = (xz)y.
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0
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OFFSET
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0,3
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more,new
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AUTHOR
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STATUS
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approved
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A362823
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Number of labeled magmas with n elements satisfying the identities (xy)y = x and (xy)z = (xz)y.
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0
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1, 1, 4, 22, 976, 19376, 7680016, 430723168, 1489656111616, 214815786486400, 6364561150037368576, 2241692646969785651456, 566719960584895502028138496, 471612192582034433034750951424, 1008512943343839231897776246546624512, 1936475539456937172034340659334701398016
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of ordered n-tuples of involutions on [n] that pairwise commute. Two involutions x,y on [n] commute if x*y = y*x.
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LINKS
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A362825
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Number of ordered triples of involutions on [n] that pairwise commute.
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0
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1, 1, 8, 22, 232, 1016, 12496, 73648, 1032032, 7586272, 118141696, 1033672256, 17668427008, 178649596672, 3313667912192, 37898019913216, 756948065453056, 9640771045925888, 205935949714235392, 2885307792776353792, 65568056040976818176
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OFFSET
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0,3
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COMMENTS
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Two involutions x,y on [n] commute if x*y = y*x.
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LINKS
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FORMULA
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E.g.f.: exp(x + 7*x^2/2 + 7*x^4/4 + x^8/8).
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PROG
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(PARI) seq(n) = {Vec(serlaplace(exp(x + 7*x^2/2 + 7*x^4/4 + x^8/8 + O(x*x^n))))}
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A362824
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Array read by antidiagonals: T(n,k) is the number of k-tuples of involutions on [n] that pairwise commute.
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0
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 1, 1, 8, 10, 10, 1, 1, 1, 16, 22, 52, 26, 1, 1, 1, 32, 46, 232, 196, 76, 1, 1, 1, 64, 94, 976, 1016, 1216, 232, 1, 1, 1, 128, 190, 4000, 4576, 12496, 5944, 764, 1, 1, 1, 256, 382, 16192, 19376, 111376, 73648, 42400, 2620, 1
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OFFSET
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0,9
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COMMENTS
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Two involutions x,y on [n] commute if x*y = y*x.
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LINKS
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FORMULA
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T(0,k) = T(1,k) = 1.
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EXAMPLE
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Array begins:
===========================================================
n/k| 0 1 2 3 4 5 6 7 ...
---+-------------------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 1 ...
2 | 1 2 4 8 16 32 64 128 ...
3 | 1 4 10 22 46 94 190 382 ...
4 | 1 10 52 232 976 4000 16192 65152 ...
5 | 1 26 196 1016 4576 19376 79696 323216 ...
6 | 1 76 1216 12496 111376 936976 7680016 62177296 ...
7 | 1 232 5944 73648 716416 6289312 52647904 430723168 ...
...
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PROG
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B(n, k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
T(n, k)={if(n==0, 1, n!*polcoef(exp(sum(j=0, min(k, logint(n, 2)), B(k, j)*x^(2^j)/2^j, O(x*x^n))), n))}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A360932
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Primes of the form H(m,k) = F(k+1)*F(m-k+2) - F(k)*F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.
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2, 3, 5, 7, 11, 13, 23, 29, 37, 41, 47, 89, 97, 103, 107, 157, 167, 173, 199, 233, 443, 521, 733, 1597, 1741, 1867, 1871, 1877, 2207, 3037, 3571, 7841, 7919, 7951, 9349, 11933, 12823, 28657, 33503, 50549, 54277, 54287, 54293, 54319, 54497, 55717, 142099
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OFFSET
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1,1
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COMMENTS
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This sequence appears in the triangle A108038 in this order (reading by rows): 3, 2, 7, 5, 11, 13, 29, 23, 47, 37, 41, 97, 107, 103, 89, 199, 157, 173, 167.
Are there infinitely many primes of the form H(m,k)?
This sequence appears within the determinant Hosoya triangle.
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LINKS
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FORMULA
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EXAMPLE
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29 is a term because it is prime and A108038(8,2) = H(8,2) = 29. Also A108038(8,7) = H(8,7) = 29.
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MATHEMATICA
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H[r_, k_] := Det[{{Fibonacci[r-k+2], Fibonacci[r-k+1]}, {Fibonacci[k], Fibonacci[k+1]}}]; DeterminantPrimes[t_, m_] := Table[If[PrimeQ[H[r, k]], H[r, k], Unevaluated[Sequence[]]], {r, t, m}, {k, 1, Ceiling[r/2]}]; ListOfPrimes[t_, m_]:= Sort[DeleteDuplicates[Flatten[DeterminantPrimes[t, m]]]]; ListOfPrimes[2, 100]
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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