A330527 Expansion of e.g.f. Sum_{k>=1} (sec(x^k) + tan(x^k) - 1).

1, 3, 8, 41, 136, 1381, 5312, 70265, 491776, 5977561, 40270592, 1021246445, 6249389056, 135671657941, 1919826163712, 36481192888145, 355897293438976, 12422529973051441, 121674189293944832, 4514836332133978325

Offset: 1

Author

Ilya Gutkovskiy, Dec 17 2019

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..449

Crossrefs

Cf. A000111, A176475, A330504, A330528.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Sum[(Sec[x^k] + Tan[x^k] - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! DivisorSum[n, If[EvenQ[#], Abs[EulerE[#]], Abs[(2^(# + 1) (2^(# + 1) - 1) BernoulliB[# + 1])/(# + 1)]]/#! &], {n, 1, 20}]

Python

from math import factorial
from itertools import accumulate
def A330527(n):
    c = a = factorial(n)
    blist = (0,1)
    for d in range(2,n+1):
        blist = tuple(accumulate(reversed(blist),initial=0))
        if n % d == 0:
            c += a*blist[-1]//factorial(d)
    return c # Chai Wah Wu, Apr 19 2023

Formula

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