A003715 Expansion of e.g.f. sin(sin(sin(x))) (odd powers only).

1, -3, 33, -731, 25857, -1311379, 89060065, -7778778091, 849264442881, -113234181108643, 18073465545032353, -3395124358886313595, 740061366713642835201, -185005977382236600650035

Offset: 0

Author

R. H. Hardin, Simon Plouffe

Keywords

• sign

Comments

{abs(a(n))} has e.g.f. sinh(sinh(sinh(x))) (odd powers only). - Jianing Song, Oct 25 2023

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

Crossrefs

Cf. A003712.

Programs

Mathematica

With[{nn = 50}, Take[CoefficientList[Series[Sin[Sin[Sin[x]]], {x, 0, nn}], x] Range[0, nn - 1]!, {2, -1, 2}]] (* Vincenzo Librandi, Apr 11 2014 *)

Maxima

a(k):=sum((sum((2^(-2*j-2*m)*(sum((2*i-2*j-1)^(2*m+1)*(-1)^(i)*binomial(2*j+1,i),i,0,(2*j+1)/2))*sum((2*i-2*m-1)^(2*k+1)*binomial(2*m+1,i)*(-1)^(k-i),i,0,(2*m+1)/2))/(2*m+1)!,m,j,k))/(2*j+1)!,j,0,k); /* Vladimir Kruchinin, Jun 10 2011 */

Formula

a(k) : =sum(j=0..k, (sum(m=j..k, (2^(-2*j-2*m)*(sum(i=0..(2*j+1)/2, (2*i-2*j-1)^(2*m+1)*(-1)^(i)*binomial(2*j+1,i)))*sum(i=0..(2*m+1)/2, (2*i-2*m-1)^(2*k+1)*binomial(2*m+1,i)*(-1)^(k-i)))/(2*m+1)!))/(2*j+1)!); [Vladimir Kruchinin, Jun 10 2011]