A331608
E.g.f.: exp(1 / (1 - sinh(x)) - 1).
Original entry on oeis.org
1, 1, 3, 14, 85, 632, 5559, 56352, 645929, 8252352, 116189291, 1786361216, 29764770941, 534082233856, 10264484355103, 210312181051392, 4575364233983057, 105310034714202112, 2556360647841415379, 65261358332774277120, 1747713179543456515749
Offset: 0
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nmax = 20; CoefficientList[Series[Exp[1/(1 - Sinh[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A006154[n_] := Sum[Sum[(-1)^j (k - 2 j)^n Binomial[k, j]/2^k, {j, 0, k}], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A006154[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
A331607
E.g.f.: exp(1 / (1 - sin(x)) - 1).
Original entry on oeis.org
1, 1, 3, 12, 61, 372, 2639, 21280, 191833, 1908688, 20750331, 244478784, 3100597333, 42088689216, 608543191559, 9332562964480, 151252803045937, 2582250195499264, 46306562212010355, 870011934425816064, 17086276243125287917
Offset: 0
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nmax = 20; CoefficientList[Series[Exp[1/(1 - Sin[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A000111[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1) (2^(n + 1) - 1) BernoulliB[n + 1])/(n + 1)]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A000111[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
A331978
E.g.f.: -log(2 - cosh(x)) (even powers only).
Original entry on oeis.org
0, 1, 4, 46, 1114, 46246, 2933074, 263817646, 31943268634, 5009616448246, 987840438629794, 239217148602642046, 69790939492563608554, 24143849395162438623046, 9772368696995766705116914, 4575221153658910691872135246, 2453303387149157947685779986874
Offset: 0
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ptan := proc(n) option remember;
if irem(n, 2) = 0 then 0 else
add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n-1, 2) fi end:
A331978 := n -> ptan(2*n - 1):
seq(A331978(n), n = 0..16); # Peter Luschny, Jun 06 2022
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nmax = 16; Table[(CoefficientList[Series[-Log[2 - Cosh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A331612
E.g.f.: exp(1 / (2 - sec(x)) - 1) (even powers only).
Original entry on oeis.org
1, 1, 14, 481, 30449, 3064306, 448104029, 89621046061, 23468873468054, 7786478152466221, 3190021872763911149, 1580829351026679822586, 931656913226081002622489, 643808850722810399312420281, 515431991397502094847830786174, 473171296200788822261644150349881
Offset: 0
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nmax = 15; Table[(CoefficientList[Series[Exp[1/(2 - Sec[x]) - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
e[0] = 1; e[n_] := e[n] = (-1)^n (1 - Sum[(-1)^j Binomial[2 n, 2 j] 3^(2 (n - j)) e[j], {j, 0, n - 1}]); A002114[n_] := e[n]/2^(2 n + 1); a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] A002114[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
With[{nn=40},Take[CoefficientList[Series[Exp[1/(2-Sec[x])-1],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Aug 08 2023 *)
Showing 1-4 of 4 results.