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A221078 E.g.f.: Sum_{n>=0} tan(n*x)^n.

%I A221078 #28 Jun 02 2022 08:14:34
%S A221078 1,1,8,164,6400,404176,37541888,4814990144,815074508800,
%T A221078 176018678814976,47223034903789568,15407438848482919424,
%U A221078 6007522256082907955200,2758698201106509138251776,1473586749521302260021198848,905915791153129699969076117504
%N A221078 E.g.f.: Sum_{n>=0} tan(n*x)^n.
%C A221078 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. A245322. - _Peter Bala_, Jun 01 2022
%H A221078 Vaclav Kotesovec, <a href="/A221078/b221078.txt">Table of n, a(n) for n = 0..224</a>
%F A221078 a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.82830192319144609189890882712268369027077465204866199572119508594067235975..., c = 0.3460492649810724519960613805096579760009441161242336020188358769124140... - _Vaclav Kotesovec_, Nov 05 2014, updated Jun 02 2022
%e A221078 E.g.f.: A(x) = 1 + x + 8*x^2/2! + 164*x^3/3! + 6400*x^4/4! + 404176*x^5/5! +...
%e A221078 where
%e A221078 A(x) = 1 + tan(x) + tan(2*x)^2 + tan(3*x)^3 + tan(4*x)^4 + tan(5*x)^5 +...
%t A221078 nmax = 20; CoefficientList[Series[1 + Sum[Tan[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, May 31 2022 *)
%t A221078 Join[{1}, Table[Sum[(-1)^((n-k)/2) * 2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Jun 01 2022 *)
%o A221078 (PARI) {a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, tan(m*X)^m); n!*polcoeff(Egf, n)}
%o A221078 for(n=0,20,print1(a(n),", ") )
%Y A221078 Cf. A122399, A195415, A220181, A221077, A221079, A245322, A224899, A249489, A338040.
%K A221078 nonn
%O A221078 0,3
%A A221078 _Paul D. Hanna_, Dec 31 2012