A003712 Expansion of e.g.f. sin(sin(x)) (odd powers only). (Formerly M2042)

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

Offset

0,2

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)

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

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 *)

Prog

(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 */

Crossrefs

Cf. A359553/A359554.

Sequence in context: A201470 A349268 A366821 * A143136 A214224 A214431

Adjacent sequences: A003709 A003710 A003711 * A003713 A003714 A003715

Keyword

sign

Author

R. H. Hardin, Simon Plouffe

Status

approved