Search: keyword:new
|
| |
|
|
A372326
|
|
Number A(n,k) of acyclic orientations of the Turán graph T(k*n,k); square array A(n,k), n>=0, k>=1, read by antidiagonals.
|
|
+0
0
|
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 14, 6, 1, 1, 1, 230, 426, 24, 1, 1, 1, 6902, 122190, 24024, 120, 1, 1, 1, 329462, 90768378, 165392664, 2170680, 720, 1, 1, 1, 22934774, 138779942046, 4154515368024, 457907248920, 287250480, 5040, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,9
|
|
|
COMMENTS
|
The Turán graph T(k*n,k) is the complete k-partite graph K_{n,...,n}.
An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, ...
1, 2, 14, 230, 6902, ...
1, 6, 426, 122190, 90768378, ...
1, 24, 24024, 165392664, 4154515368024, ...
1, 120, 2170680, 457907248920, 495810323060597880, ...
|
|
|
MAPLE
|
g:= proc(n) option remember; `if`(n=0, 1, add(
expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
end:
T:= proc(n, k) option remember; local q, l, b; q, l, b:= -1, [k$n, 0],
proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*
(q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))
end; abs(b(0, nops(l)))
end:
seq(seq(T(n, d-n), n=0..d), d=0..10);
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A372228
|
|
a(n) is the largest prime factor of n^n + n.
|
|
+0
0
|
|
|
|
2, 3, 5, 13, 313, 101, 181, 5419, 21523361, 52579, 212601841, 57154490053, 815702161, 100621, 4454215139669, 4562284561, 52548582913, 1895634885375961, 211573, 2272727294381, 415710882920521, 9299179, 1853387306082786629, 22496867303759173834520497
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
MATHEMATICA
|
Table[f = FactorInteger[n^n + n]; f[[Length[f]]][[1]], {n, 1, 25}] (* Vaclav Kotesovec, Apr 26 2024 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,new
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A372229
|
|
a(n) is the largest prime factor of n^n - n.
|
|
+0
0
|
|
|
|
2, 3, 7, 13, 311, 43, 337, 193, 333667, 13421, 266981089, 28393, 29914249171, 10678711, 1321, 184417, 7563707819165039903, 236377, 192696104561, 920421641, 12271836836138419, 39700406579747, 58769065453824529, 152587500001, 4315817869647001, 797161
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
2,1
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
MAPLE
|
pf := n -> NumberTheory:-PrimeFactors(n): a := n -> max(pf(n^n - n));
|
|
|
MATHEMATICA
|
Table[f = FactorInteger[n^n-n]; f[[Length[f]]][[1]], {n, 2, 25}] (* Vaclav Kotesovec, Apr 26 2024 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,new
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A371936
|
|
Decimal expansion of Sum_{k>=0} (-1)^k / (2*(k+1)*(2*k+1)!).
|
|
+0
0
|
|
|
|
4, 5, 9, 6, 9, 7, 6, 9, 4, 1, 3, 1, 8, 6, 0, 2, 8, 2, 5, 9, 9, 0, 6, 3, 3, 9, 2, 5, 5, 7, 0, 2, 3, 3, 9, 6, 2, 6, 7, 6, 8, 9, 5, 7, 9, 3, 8, 2, 0, 7, 7, 7, 7, 2, 3, 2, 9, 9, 0, 2, 7, 4, 4, 6, 1, 8, 8, 9, 9, 6, 0, 5, 2, 2, 5, 5, 2, 8, 2, 3, 5, 4, 8, 2, 0, 4
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
0.459697694131860282599063392557023396267689579382077772...
|
|
|
MATHEMATICA
|
s = N[Sum[(-1)^k/(2(k + 1) (2 k + 1)!), {k, 0, Infinity}], 120]
First[RealDigits[s]]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A371945
|
|
Decimal expansion of Sum_{k>=0} (-1)^k / (k^3 + 1).
|
|
+0
0
|
|
|
|
5, 8, 5, 6, 9, 8, 8, 6, 1, 9, 4, 9, 7, 9, 1, 8, 8, 6, 4, 5, 3, 3, 2, 7, 0, 4, 6, 9, 5, 9, 1, 8, 6, 1, 5, 3, 9, 7, 5, 3, 6, 3, 0, 2, 1, 2, 8, 6, 9, 4, 9, 2, 8, 3, 7, 4, 7, 5, 2, 7, 3, 3, 2, 7, 7, 8, 0, 9, 0, 1, 4, 0, 7, 0, 0, 9, 4, 3, 8, 5, 6, 8, 2, 9, 9, 7
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
0.58569886194979188645332704695918615397536302...
|
|
|
MATHEMATICA
|
s = Chop[N[Sum[(-1)^k/(k^3 + 1), {k, 0, Infinity}], 120]]
First[RealDigits[s]]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A371935
|
|
Decimal expansion of Sum_{k>=0} (-1)^k / ((k+1)*(2*k+1)!).
|
|
+0
0
|
|
|
|
9, 1, 9, 3, 9, 5, 3, 8, 8, 2, 6, 3, 7, 2, 0, 5, 6, 5, 1, 9, 8, 1, 2, 6, 7, 8, 5, 1, 1, 4, 0, 4, 6, 7, 9, 2, 5, 3, 5, 3, 7, 9, 1, 5, 8, 7, 6, 4, 1, 5, 5, 5, 4, 4, 6, 5, 9, 8, 0, 5, 4, 8, 9, 2, 3, 7, 7, 9, 9, 2, 1, 0, 4, 5, 1, 0, 5, 6, 4, 7, 0, 9, 6, 4, 0, 9
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
LINKS
|
|
|
|
FORMULA
|
Equals 2*(1 - cos(1)) = 2 * A371936.
|
|
|
EXAMPLE
|
0.91939538826372056519812678511404679253...
|
|
|
MATHEMATICA
|
s = N[Sum[(-1)^k/((k + 1) (2 k + 1)!), {k, 0, Infinity}], 120]
First[RealDigits[s]]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A372314
|
|
Determinant of the matrix [Jacobi(i^2 + 3*i*j + 2*j^2, 2*n + 1)]_{1 < i, j < 2*n}, where Jacobi(a, m) denotes the Jacobi symbol (a / m).
|
|
+0
0
|
|
|
|
1, 3, 0, 125, -1215, 0, 0, 9126441, 0, -187590821, 0, 0, 20686753425, 0, 0, 0, 9224101117395305225, 0, 881852208012283730302080, 624391710361368134976, 0, -3428714319207136609529065, 0, 0, 3878246452353765171209988566241, 0, 0, 4308304210666498856284267223158421
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
2,2
|
|
|
COMMENTS
|
Conjecture 1: Let n be any positive integer.
(i) If a(2*n) is nonzero, then 4*n + 1 is a sum of two squares.
(ii) a(2*n + 1) is divisible by phi(4*n + 3)/2, where phi is Euler's totient function. If n is even, then a(2*n + 1)/(phi(4*n + 3)/2) is a square. This has been verified for n = 2..1000.
For any odd integer n > 3 and integers c and d, we introduce the notation: {c,d}_n = det[Jacobi(i^2 + c*i*j + d*j^2, n)]_{1 < i, j < n-1}.
The following conjecture is similar to Conjecture 1.
Conjecture 2: (1) {2, 2}_p = 0 for any prime p == 13,19 (mod 24), and {2, 2}_p == 0 (mod p) for any prime p == 17,23 (mod 24).
(2) If n == 5 (mod 8), then {4, 2}_n = 0. If n == 5 (mod 12), then {3, 3}_n = 0.
(3) If n == 5 (mod 12) and n is a sum of two squares, then {10, 9}_n = 0. Also, {10, 9}_p == 0 (mod p) for any prime p == 11 (mod 12).
(4) {8, 18}_p == 0 (mod p^2) for any prime p == 19 (mod 24), and {8,18}_p == 0 (mod p) for any prime p == 23 (mod 24). If n == 13,17 (mod 24) and n is a sum of two squares, then {8, 18}_n = 0.
We have verified Conjecture 2 for p or n smaller than 2000.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
a(2) = 1 since the determinant of the matrix [Jacobi(i^2 + 3*i*j + 2*j^2, 5)]_{1 < i, j < 2*2} = [1,0; 0,1] is 1.
|
|
|
MATHEMATICA
|
a[n_]:=a[n]=Det[Table[JacobiSymbol[i^2+3*i*j+2*j^2, 2n+1], {i, 2, 2n-1}, {j, 2, 2n-1}]];
tab={}; Do[tab=Append[tab, a[n]], {n, 2, 29}]; Print[tab]
|
|
|
PROG
|
(PARI) f(i, j) = i^2 + 3*i*j + 2*j^2;
a(n) = matdet(matrix(2*n-2, 2*n-2, i, j, kronecker(f(i+1, j+1), 2*n+1)));
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign,new
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A372315
|
|
Expansion of e.g.f. exp( x - LambertW(-2*x)/2 ).
|
|
+0
0
|
|
|
|
1, 2, 8, 68, 960, 18832, 471136, 14324480, 512733696, 21119803136, 984029612544, 51169331031040, 2937675286583296, 184560174104465408, 12594824112085327872, 927757127285523243008, 73369903633161123397632, 6200198958236463387836416
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k / (1-x)^(k+1).
|
|
|
PROG
|
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-2*x)/2)))
(PARI) a(n) = sum(k=0, n, (2*k+1)^(k-1)*binomial(n, k));
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,new
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A372316
|
|
Expansion of e.g.f. exp( x - LambertW(-3*x)/3 ).
|
|
+0
0
|
|
|
|
1, 2, 10, 125, 2644, 77597, 2904382, 132169403, 7083715240, 437031850841, 30506442905194, 2377038378159359, 204521399708464252, 19259006462435865413, 1970114326513629358654, 217556451608123850352523, 25794252755430105917806288, 3268152272130255473300883377
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (3*k+1)^(k-1) * x^k / (1-x)^(k+1).
|
|
|
PROG
|
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x)/3)))
(PARI) a(n) = sum(k=0, n, (3*k+1)^(k-1)*binomial(n, k));
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,new
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A372321
|
|
Expansion of e.g.f. -exp( x + LambertW(-3*x)/3 ).
|
|
+0
0
|
|
|
|
-1, 0, 6, 81, 1620, 45765, 1671678, 74794671, 3958829640, 241898775273, 16756621904970, 1297547591499819, 111065107263415308, 10412999996499836541, 1061234184094567585326, 116812280111404106348415, 13810631408232372091755792, 1745470697932523785587735249
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} (3*k-1)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=0} (3*k-1)^(k-1) * x^k / (1-x)^(k+1).
|
|
|
PROG
|
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x+lambertw(-3*x)/3)))
(PARI) a(n) = sum(k=0, n, (3*k-1)^(k-1)*binomial(n, k));
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign,new
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.122 seconds
|
