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E.g.f.: Sum_{n>=0} sin(n*x)^n.
8

%I #36 Jun 02 2022 01:58:03

%S 1,1,8,161,6016,360421,31628288,3823725821,609263681536,

%T 123729353398441,31195066498285568,9560281195915697081,

%U 3500145542231863853056,1508772905238685631514061,756360258034794813559144448,436312320288025061112662937941,286966475921556619941746443288576

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

%C 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

%H Vaclav Kotesovec, <a href="/A245322/b245322.txt">Table of n, a(n) for n = 0..200</a>

%F 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

%t nmax=20; Flatten[{1,Rest[CoefficientList[Series[Sum[Sin[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]}]

%t 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}]}]

%o (PARI) {a(n)=n!*polcoeff(sum(k=0, n, sin(k*x+x*O(x^n))^k), n)}

%o for(n=0, 20, print1(a(n), ", "))

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

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Nov 05 2014