A176474
From the coefficients (times n!) of the expansion of the Sum_{k>=1} cos(x^k).
Original entry on oeis.org
0, 1, 11, 361, 18479, 1814401, 220207679, 43589145601, 9589093113599, 3210079038566401, 1115080757523532799, 562000363888803840001, 285218528618534141030399, 201645730563302817792000001
Offset: 1
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Rest@ Union[ -Range[0, 26]! CoefficientList[ Series[ Sum[ Cos[x^n], {n, 35}], {x, 0, 27}], x]]
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{ my(N=33, x='x+O('x^N)); v = Vec(serlaplace(-sum(k=1,N,cos(x^k)))); vector(#v\2-1, k, v[2*k+1])} \\ \\ Joerg Arndt, Apr 21 2017
A176473
The coefficients (times n!) of the expansion of the sum {k=1 to Inf.} of Sin(x^n).
Original entry on oeis.org
0, 1, 2, 5, 24, 121, 600, 5039, 40320, 302401, 3659040, 39916799, 399168000, 6227020801, 87160993920, 1100625926399, 20922789888000, 355687428096001, 5335329064665600, 121645100408831999, 2453176191578112000
Offset: 1
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Range[0, 25]! CoefficientList[ Series[ Sum[ Sin[x^n], {n, 35}], {x, 0, 25}], x]
A330504
Expansion of e.g.f. Sum_{k>=1} tanh(x^k).
Original entry on oeis.org
1, 2, 4, 24, 136, 480, 4768, 40320, 249856, 4112640, 39563008, 319334400, 6249389056, 82473431040, 1044235737088, 20922789888000, 355897293438976, 4408265775513600, 121616011523719168, 2757288942600192000, 31308290669925892096
Offset: 1
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nmax = 21; CoefficientList[Series[Sum[Tanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
A155585[n_] := Sum[StirlingS2[n, k] (-2)^(n - k) k!, {k, 0, n}]; a[n_] := n! DivisorSum[n, A155585[#]/#! &]; Table[a[n], {n, 1, 21}]
Table[n! DivisorSum[n, 2^(# + 1) (2^(# + 1) - 1) BernoulliB[# + 1]/(# + 1)! &, OddQ[#] &], {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 1-5 of 5 results.