A330505
Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).
Original entry on oeis.org
1, 2, 8, 24, 144, 960, 5760, 40320, 524160, 4354560, 43545600, 638668800, 6706022400, 99632332800, 2092278988800, 20922789888000, 376610217984000, 9247873130496000, 128047474114560000, 2919482409811968000, 77852864261652480000
Offset: 1
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nmax = 21; CoefficientList[Series[Sum[ArcTanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
nmax = 21; CoefficientList[Series[-Log[EllipticTheta[4, 0, x]]/2, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, # &, OddQ[n/#] &], {n, 1, 21}]
A330527
Expansion of e.g.f. Sum_{k>=1} (sec(x^k) + tan(x^k) - 1).
Original entry on oeis.org
1, 3, 8, 41, 136, 1381, 5312, 70265, 491776, 5977561, 40270592, 1021246445, 6249389056, 135671657941, 1919826163712, 36481192888145, 355897293438976, 12422529973051441, 121674189293944832, 4514836332133978325
Offset: 1
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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}]
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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
A330511
Expansion of e.g.f. Sum_{k>=1} arctan(x^k).
Original entry on oeis.org
1, 2, 4, 24, 144, 480, 4320, 40320, 282240, 4354560, 36288000, 319334400, 6706022400, 74724249600, 1046139494400, 20922789888000, 376610217984000, 4979623993344000, 115242726703104000, 2919482409811968000, 29194824098119680000
Offset: 1
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nmax = 21; CoefficientList[Series[Sum[ArcTan[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, (-1)^((n/# - 1)/2) # &, OddQ[n/#] &], {n, 1, 21}]
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a(n) = (n-1)!*sumdiv(n, d, if (n/d % 2, (-1)^((n/d - 1)/2)*d)); \\ Michel Marcus, Dec 17 2019
Showing 1-3 of 3 results.