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}]
A330506
Expansion of e.g.f. Sum_{k>=1} arcsin(x^k).
Original entry on oeis.org
1, 2, 7, 24, 129, 840, 5265, 40320, 434385, 3900960, 40809825, 558835200, 6335076825, 91070179200, 1641957141825, 20922789888000, 359796258446625, 7663952552256000, 122832552380162625, 2615369658789888000, 62315614994643635625
Offset: 1
-
nmax = 21; CoefficientList[Series[Sum[ArcSin[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! DivisorSum[n, ((# - 2)!!)^2/#! &, OddQ[#] &], {n, 1, 21}]
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
-
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
A330512
Expansion of e.g.f. Sum_{k>=1} arcsinh(x^k).
Original entry on oeis.org
1, 2, 5, 24, 129, 600, 4815, 40320, 313425, 3900960, 39023775, 399168000, 6335076825, 83286403200, 1169542749375, 20922789888000, 359796258446625, 5529827983680000, 120457648437501375, 2615369658789888000, 40723609672075955625
Offset: 1
-
nmax = 21; CoefficientList[Series[Sum[ArcSinh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! DivisorSum[n, (-1)^((# - 1)/2) ((# - 2)!!)^2/#! &, OddQ[#] &], {n, 1, 21}]
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