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Expansion of e.g.f. Sum_{k>=1} (sec(x^k) + tan(x^k) - 1).
2

%I #13 Apr 20 2023 02:10:36

%S 1,3,8,41,136,1381,5312,70265,491776,5977561,40270592,1021246445,

%T 6249389056,135671657941,1919826163712,36481192888145,355897293438976,

%U 12422529973051441,121674189293944832,4514836332133978325

%N Expansion of e.g.f. Sum_{k>=1} (sec(x^k) + tan(x^k) - 1).

%H Chai Wah Wu, <a href="/A330527/b330527.txt">Table of n, a(n) for n = 1..449</a>

%F a(n) = n! * Sum_{d|n} A000111(d) / d!.

%t nmax = 20; CoefficientList[Series[Sum[(Sec[x^k] + Tan[x^k] - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%t Table[n! DivisorSum[n, If[EvenQ[#], Abs[EulerE[#]], Abs[(2^(# + 1) (2^(# + 1) - 1) BernoulliB[# + 1])/(# + 1)]]/#! &], {n, 1, 20}]

%o (Python)

%o from math import factorial

%o from itertools import accumulate

%o def A330527(n):

%o c = a = factorial(n)

%o blist = (0,1)

%o for d in range(2,n+1):

%o blist = tuple(accumulate(reversed(blist),initial=0))

%o if n % d == 0:

%o c += a*blist[-1]//factorial(d)

%o return c # _Chai Wah Wu_, Apr 19 2023

%Y Cf. A000111, A176475, A330504, A330528.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Dec 17 2019