A003712 Expansion of e.g.f. sin(sin(x)) (odd powers only).

1, -2, 12, -128, 1872, -37600, 990784, -32333824, 1272660224, -59527313920, 3252626013184, -204354574172160, 14594815769038848, -1174376539738169344, 105595092426069327872, -10530693390637550272512

Offset: 0

Author

R. H. Hardin, Simon Plouffe

Keywords

• sign

Comments

abs(a(n)) has e.g.f. sinh(sinh(x)) (odd powers only).

abs(a(n)) is the number of partitions of the set {1, 2, ..., 2*n-1} into an odd number of blocks, each containing an odd number of elements. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 23 2004

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 6th line of table.
• 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 (first 50 terms from T. D. Noe)

Crossrefs

Cf. A359553/A359554.

Programs

Mathematica

With[{max = 50}, Take[CoefficientList[Series[Sin[Sin[x]], {x, 0, max}], x] Range[0, max - 1]!, {2, -1, 2}]] (* Vincenzo Librandi, Apr 11 2014 *)
Table[Sum[(-1)^(m + n) (1 + 2k - 2m)^(2n + 1)/(4^k (1 + 2k - m)! m!), {k, 0, n}, {m, 0, k + 1/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)

Maxima

a(n):=sum((sum((2*i-2*j+1)^(2*n+1)*binomial(2*j-1,i)*(-1)^(n-i-1),i,0,(2*j-1)/2)/(4^(j-1)*(2*j-1)!)),j,1,n+1); /* Vladimir Kruchinin, Jun 09 2011 */

Formula

a(n) = Sum_{j=1..n+1} (1/(4^(j-1)*(2*j-1)!)) * Sum_{i=0..(2*j-1)/2} (2*i-2*j+1)^(2*n+1) * binomial(2*j-1,i)*(-1)^(n-i-1). - Vladimir Kruchinin, Jun 09 2011