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Displaying 1-10 of 219 results found. page 1 2 3 4 5 6 7 8 9 10 ... 22
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A363417 a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A119259(k)*x^k/k ). +0
0
1, 3, 35, 462, 6435, 92378, 1352078, 20058300, 300540195, 4537567650, 68923264410, 1052049481860, 16123801841550, 247959266474052, 3824345300380220, 59132290782430712, 916312070471295267, 14226520737620288370, 221256270138418389602, 3446310324346630677300 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Note that E(x) = exp( Sum_{k >= 1} A119259(k)*x^k/k ) = 1 + 3*x + 13*x^2 + 67*x^3 + ... is the g.f. of A062992
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 5 and positive integers n and r. The same supercongruences are also conjectured to hold for A119259.
More generally, we inductively define a family of sequences {a(i,n) : n >= 0}, i >= 0, by setting a(0,n) = A119259(n) and, for i >= 1, a(i,n) = [x^n] ( exp( Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. In this notation the present sequence is {a(1,n)}. We conjecture that the each sequence {a(i,n) : n >= 0}, i >= 1, satisfies the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5, and positive integers n and r. Cf. A362728.
LINKS
Table of n, a(n) for n=0..19.
FORMULA
a(n) = [x^n] exp( n*Sum_{k >= 1} A119259(k)*x^k/k ).
EXAMPLE
Examples of supercongruences:
a(11) - a(1) = 1052049481860 - 3 = (3^2)*(11^3)*6337*13859 == 0 (mod 11^3).
a(3*5) - a(3) = 59132290782430712 - 462 = 2*(5^3)*7*11*191*16082760803 == 0 (mod 5^3).
MAPLE
A119259 := proc(n) option remember; add(binomial(2*n, k)*binomial(2*n-k-1, n-k), k = 0..n) end:
E(n, x) := series( exp(n*add(A119259(k)*x^k/k, k = 1..20)), x, 21):
seq(coeftayl(E(n, x), x = 0, n), n = 0..20);
CROSSREFS
Cf. A062992, A119259, A362724 - A362733.
KEYWORD
nonn,easy,new
AUTHOR
Peter Bala, Jun 08 2023
STATUS
approved
A363543 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (3^k + A(x^k)) * x^k/k ). +0
0
1, 4, 7, 23, 69, 234, 826, 3000, 11168, 42313, 162829, 634052, 2495051, 9903761, 39612048, 159481988, 645833656, 2628829700, 10749777653, 44139474552, 181916530895, 752288709592, 3120574260606, 12981015704961, 54138655342763, 226330448292140 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..25.
FORMULA
A(x) = Sum_{k>=0} a(k) * x^k = (1+3*x) * Product_{k>=0} (1+x^(k+1))^a(k).
a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( (-3)^k + Sum_{d|k} (-1)^(k/d) * d * a(d-1) ) * a(n-k).
PROG
(PARI) seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (-1)^(k+1)*(3^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);
CROSSREFS
Cf. A038075, A363542.
Cf. A363509, A363541.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jun 09 2023
STATUS
approved
A363542 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (2^k + A(x^k)) * x^k/k ). +0
0
1, 3, 5, 14, 38, 114, 360, 1166, 3872, 13094, 44961, 156244, 548636, 1943333, 6935817, 24917586, 90039163, 327029681, 1193258619, 4371901789, 16077606949, 59325057056, 219579151797, 815017718383, 3032959638204, 11313632991360, 42295634914403 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..26.
FORMULA
A(x) = Sum_{k>=0} a(k) * x^k = (1+2*x) * Product_{k>=0} (1+x^(k+1))^a(k).
a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( (-2)^k + Sum_{d|k} (-1)^(k/d) * d * a(d-1) ) * a(n-k).
PROG
(PARI) seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (-1)^(k+1)*(2^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);
CROSSREFS
Cf. A038075, A363543.
Cf. A362389.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jun 09 2023
STATUS
approved
A363545 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - 2*x^k)) ). +0
0
1, 1, 4, 14, 54, 206, 823, 3312, 13619, 56643, 238569, 1014443, 4352038, 18809992, 81843021, 358186642, 1575810191, 6965004499, 30914431131, 137736012285, 615785575785, 2761693248028, 12421390811559, 56016050571825, 253228531426237 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..24.
FORMULA
A(x) = (1 - 2*x) * B(x) where B(x) is the g.f. of A362389.
a(n) = A362389(n) - 2*A362389(n-1) for n > 0.
PROG
(PARI) seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, x^k)*x^k/(k*(1-2*x^k)))+x*O(x^n))); Vec(A);
CROSSREFS
Cf. A052855, A363546.
Cf. A362389.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jun 09 2023
STATUS
approved
A362389 G.f. satisfies A(x) = exp( Sum_{k>=1} (2^k + A(x^k)) * x^k/k ). +0
0
1, 3, 10, 34, 122, 450, 1723, 6758, 27135, 110913, 460395, 1935233, 8222504, 35255000, 152353021, 662892684, 2901595559, 12768195617, 56450822365, 250637657015, 1117060889815, 4995815027658, 22413020866875, 100842092305575, 454912716037387 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..24.
FORMULA
A(x) = B(x)/(1 - 2*x) where B(x) is the g.f. of A363545.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-2*x) * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 2^k + Sum_{d|k} d * a(d-1) ) * a(n-k).
PROG
(PARI) seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (2^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);
CROSSREFS
Cf. A001678, A036249, A363541.
Cf. A029857, A363542, A363545.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jun 09 2023
STATUS
approved
A363541 G.f. satisfies A(x) = exp( Sum_{k>=1} (3^k + A(x^k)) * x^k/k ). +0
0
1, 4, 17, 73, 324, 1469, 6838, 32490, 157398, 775010, 3870690, 19567202, 99957231, 515250057, 2676884745, 14002926871, 73693381322, 389904743248, 2072794614996, 11066421965311, 59310040841395, 318978744562253, 1720962766007827 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..22.
FORMULA
A(x) = B(x)/(1 - 3*x) where B(x) is the g.f. of A363546.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-3*x) * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 3^k + Sum_{d|k} d * a(d-1) ) * a(n-k).
PROG
(PARI) seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (3^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);
CROSSREFS
Cf. A001678, A036249, A362389.
Cf. A363507, A363543, A363546.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jun 09 2023
STATUS
approved
A363546 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - 3*x^k)) ). +0
0
1, 1, 5, 22, 105, 497, 2431, 11976, 59928, 302816, 1545660, 7955132, 41255625, 215378364, 1131134574, 5972272636, 31684600709, 168824599282, 903080385252, 4848038120323, 26110774945462, 141048622038068, 764026532321068, 4149020129689451 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..23.
FORMULA
A(x) = (1 - 3*x) * B(x) where B(x) is the g.f. of A363541.
a(n) = A363541(n) - 3*A363541(n-1) for n > 0.
PROG
(PARI) seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, x^k)*x^k/(k*(1-3*x^k)))+x*O(x^n))); Vec(A);
CROSSREFS
Cf. A052855, A363545.
Cf. A363541.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Jun 09 2023
STATUS
approved
A363141 Expansion of g.f. A(x) satisfying 1/x = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 1, a(1) = 1. +0
0
1, 1, 0, 2, 3, 11, 23, 76, 188, 575, 1587, 4732, 13714, 40993, 121787, 367100, 1107371, 3367412, 10267404, 31468401, 96734992, 298488537, 923587457, 2866241029, 8916951360, 27808418089, 86910042122, 272180834822, 854004007736, 2684311988984, 8451232727631 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1/x = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
(2) 1/x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
(3) A(x)/x = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
(4) A(x)/x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).
a(n) ~ c * d^n / n^(3/2), where d = 3.3064656105288278... and c = 0.3845291573508... - Vaclav Kotesovec, Jun 09 2023
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^3 + 3*x^4 + 11*x^5 + 23*x^6 + 76*x^7 + 188*x^8 + 575*x^9 + 1587*x^10 + 4732*x^11 + 13714*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-1 + x*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-2); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A357227, A363312, A363313, A363314, A363315.
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jun 09 2023
STATUS
approved
A363115 Expansion of e.g.f. log(1 - log( sqrt(1-2*x) )). +0
0
0, 1, 1, 4, 22, 168, 1616, 18800, 256432, 4012288, 70825344, 1392214272, 30157260288, 713680180224, 18319344307200, 506934586748928, 15043324048398336, 476540007615725568, 16050059458251915264, 572710950848334200832, 21582629580640554123264, 856552661738538476765184 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..21.
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = log(1 - (1/2)*log(1-2*x)).
(2) a(n) = (-1)^(n-1) * Sum_{k=1..n} 2^(n-k) * (k-1)! * Stirling1(n, k) for n > 0.
(3) a(n) = 2^(n-1)*(n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * 2^(k-1) * a(n-k) for n > 0.
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 22*x^4/4! + 168*x^5/5! + 1616*x^6/6! + 18800*x^7/7! + 256432*x^8/8! + 4012288*x^9/9! + ...
where
exp(A(x)) = 1 + x + 2*x^2/2 + 4*x^3/3 + 8*x^4/4 + 16*x^5/5 + ... + 2^(n-1)*x^n/n + ...
PROG
(PARI) {a(n) = n!*polcoeff( log((1 - log(sqrt(1-2*x +x*O(x^n))))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = (-1)^(n-1) * sum(k=1, n, 2^(n-k) * (k-1)! * stirling(n, k, 1) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if (n<1, 0, 2^(n-1)*(n-1)! - sum(k=1, n-1, binomial(n-1, k)*(k-1)! * 2^(k-1) * a(n-k)))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Cf. A089064, A003713, A363116.
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jun 09 2023
STATUS
approved
A363116 Expansion of e.g.f. log(1 - (1/3)*log(1-3*x)). +0
0
0, 1, 2, 11, 93, 1068, 15486, 271206, 5566086, 130982328, 3476230344, 102709363392, 3343387479840, 118880973126576, 4584247231485312, 190548125567321328, 8492669888285758896, 404023626910206388224, 20434095445804056842112, 1094849162137482139541376 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..19.
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = log(1 - (1/3)*log(1-3*x)).
(2) a(n) = (-1)^(n-1) * Sum_{k=1..n} 3^(n-k) * (k-1)! * Stirling1(n, k) for n > 0.
(3) a(n) = 3^(n-1)*(n-1)! - Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * 3^(k-1) * a(n-k) for n > 0.
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 93*x^4/4! + 1068*x^5/5! + 15486*x^6/6! + 271206*x^7/7! + 5566086*x^8/8! + 130982328*x^9/9! + ...
where
exp(A(x)) = 1 + x + 3*x^2/2 + 9*x^3/3 + 27*x^4/4 + 81*x^5/5 + ... + 3^(n-1)*x^n/n + ...
PROG
(PARI) {a(n) = n!*polcoeff( log((1 - (1/3)*log(1-3*x +x*O(x^n) ))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = (-1)^(n-1) * sum(k=1, n, 3^(n-k) * (k-1)! * stirling(n, k, 1) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if (n<1, 0, 3^(n-1)*(n-1)! - sum(k=1, n-1, binomial(n-1, k)*(k-1)! * 3^(k-1) * a(n-k)))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Cf. A089064, A003713, A363115.
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jun 09 2023
STATUS
approved
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Last modified June 9 11:22 EDT 2023. Contains 363178 sequences. (Running on oeis4.)