A003717 Expansion of e.g.f. sin(tanh(x)) (odd powers only). (Formerly M3143)

1, -3, 37, -959, 41641, -2693691, 241586893, -28607094455, 4315903789009, -807258131578995, 183184249105857781, -49548882107764546223, 15742588857552887269753, -5802682207845642276301995, 2455154569684852694166018205, -1181492983103332335195255665063, 641493310427219875145163434181025

Offset

0,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Peter Bala, The method of Graves for computing inverse functions.

Formula

a(n) = (-1)^(n-1)*b(2*n-1), where b(n) = Sum_{k = 1..n} (1-(-1)^k)/k!*((-1)^(n-k)+1)*Sum_{j = k..n} binomial(j-1,k-1)*j!*2^(n-j-2)*(-1)^((n+k)/2+j)*Stirling2(n,j). - Vladimir Kruchinin, Apr 21 2011

For n >= 1, set d(n, x) = (1 - x^2)*d/dx(d(n-1, x)) with d(0, x) = sin(x). Then a(n) = d(2*n-1, 0). - Peter Bala, Jan 27 2026

Maple

d := proc(n, x) option remember; if n = 0 then sin(x) else simplify( (1 - x^2)*diff( d(n-1, x), x) ) end if end proc:
seq( eval(d(2*n-1, x), x = 0), n = 1..20 ); # Peter Bala, Jan 27 2026

Mathematica

Sin[ Tanh[ x ] ] (* Odd Part *)
With[{nn = 60}, Take[CoefficientList[Series[Sin[Tanh[x]], {x, 0, nn}], x] Range[0, nn - 1]!, {2, -1, 2}]] (* Vincenzo Librandi, Apr 11 2014 *)

Prog

(Maxima)
a(n):=(-1)^(n-1)*b(2*n-1);
b(n):=sum((1-(-1)^k)/k!*((-1)^(n-k)+1)*sum(binomial(j-1, k-1)*j!*2^(n-j-2)*(-1)^((n+k)/2+j)*stirling2(n, j), j, k, n), k, 1, n); /* Vladimir Kruchinin, Apr 21 2011 */

Crossrefs

Cf. A003706.

Sequence in context: A382030 A331656 A382016 * A354020 A201697 A274308

Adjacent sequences: A003714 A003715 A003716 * A003718 A003719 A003720

Keyword

sign,easy

Author

R. H. Hardin, Simon Plouffe

Extensions

Name edited by Michel Marcus, Jan 28 2018

Status

approved