A245322 E.g.f.: Sum_{n>=0} sin(n*x)^n.

1, 1, 8, 161, 6016, 360421, 31628288, 3823725821, 609263681536, 123729353398441, 31195066498285568, 9560281195915697081, 3500145542231863853056, 1508772905238685631514061, 756360258034794813559144448, 436312320288025061112662937941, 286966475921556619941746443288576

Offset

0,3

Comments

It appears that for n >= 1, a(2*n) is even and a(2*n-1) is odd. Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic. If p = 4*m + 1 the period appears to be p - 1, while if p = 4*m + 3 the period appears to be 2*(p - 1). Cf. A224899 and A221078. - Peter Bala, May 31 2022

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.6508143537621057095493599669955786931108630276472035393383790812849064745..., c = 0.447880926276318254580767843378566025547642779941081708311676940459098... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

Mathematica

nmax=20; Flatten[{1, Rest[CoefficientList[Series[Sum[Sin[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]}]
Flatten[{1, Table[Sum[(-1)^k * (n-2*k)^n * 2^(2*k-n) * Sum[Binomial[n-2*k, j] * (-1)^j * (n-2*k-2*j)^n, {j, 0, n-2*k}], {k, 0, n/2}], {n, 1, 20}]}]

Prog

(PARI) {a(n)=n!*polcoeff(sum(k=0, n, sin(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A122399, A195415, A220181, A221077, A221078, A224899, A249489, A338040.

Sequence in context: A195607 A064755 A140337 * A305524 A326099 A171461

Adjacent sequences: A245319 A245320 A245321 * A245323 A245324 A245325

Keyword

nonn

Author

Vaclav Kotesovec, Nov 05 2014

Status

approved