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A003712 E.g.f. sin(sin(x)) (odd powers only).
(Formerly M2042)
5

%I M2042

%S 1,-2,12,-128,1872,-37600,990784,-32333824,1272660224,-59527313920,

%T 3252626013184,-204354574172160,14594815769038848,

%U -1174376539738169344,105595092426069327872,-10530693390637550272512

%N E.g.f. sin(sin(x)) (odd powers only).

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

%C 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

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 6th line of table.

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

%H T. D. Noe and Vincenzo Librandi, <a href="/A003712/b003712.txt">Table of n, a(n) for n = 0..100</a> (first 50 terms from T. D. Noe)

%F a(n) = sum(j=1..n+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))/(4^(j-1)*(2*j-1)!))). [_Vladimir Kruchinin_, Jun 09 2011]

%t 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 *)

%t 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 *)

%o (Maxima)

%o 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 */

%K sign

%O 0,2

%A _R. H. Hardin_, _Simon Plouffe_

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Last modified February 27 04:09 EST 2020. Contains 332299 sequences. (Running on oeis4.)