A003722 Expansion of e.g.f. sin(sinh(x)) (odd powers only). (Formerly M4541)

1, 0, -8, -56, 64, 12672, 309376, 2917888, -163782656, -12716052480, -495644917760, -4004259037184, 1359174582304768, 153146435763437568, 9620207941273255936, 142966118908253962240, -62984506636368191946752, -11370531559240021108064256, -1167313081294818332899278848

Offset

0,3

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

Formula

a(n) = Sum_{j=1..n+1} (Sum_{i=0..2*j-1} (-1)^(i+j-1)*(2*j-1-2*i)^(2*n+1)*binomial(2*j-1,i))/(2^(2*j-1)*(2*j-1)!). - Vladimir Kruchinin, Jun 09 2011

Conjecture: a(n) = -2 * Sum_{k=1..oo} (-1)^k * (2*k - 1)^(2*n + 1) * BesselI(2*k - 1, 1). - Velin Yanev, Feb 20 2026

Mathematica

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

Prog

(Maxima)
a(n):=sum((sum((-1)^(i+j-1)*(2*j-1-2*i)^(2*n+1)*binomial(2*j-1, i), i, 0, 2*j-1))/(2^(2*j-1)*(2*j-1)!), j, 1, n+1); /* Vladimir Kruchinin, Jun 09 2011 */
(PARI) my(x='x+O('x^50), v=Vec(serlaplace(sin(sinh(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Feb 20 2026

Crossrefs

Sequence in context: A264644 A144748 A160429 * A364124 A044146 A118772

Adjacent sequences: A003719 A003720 A003721 * A003723 A003724 A003725

Keyword

sign

Author

R. H. Hardin, Simon Plouffe

Status

approved