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A003709
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E.g.f. cos(sin(x)) (even powers only).
(Formerly M3986)
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5
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1, -1, 5, -37, 457, -8169, 188685, -5497741, 197920145, -8541537105, 432381471509, -25340238127989, 1699894200469849, -129076687233903673, 10989863562589199389, -1041327644107761435101, 109095160722852951673633, -12561989444137938396142753
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OFFSET
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0,3
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COMMENTS
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|a(n)| is the number of ways to partition the set {1,2,...,2n} into an even number of odd size blocks. - Geoffrey Critzer, Apr 11 2010
Unsigned sequence has e.g.f. cosh(sinh(x)) (even powers only).
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 226, 8th line of table.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..50
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FORMULA
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a(n) = sum(j=0..n, (2^(2*j+1)*sum(i=0..(n-j), (i-n+j)^(2*n)*binomial((2*n-2*j),i)*(-1)^(n-i))/(2*n-2*j)!)), n>0, a(1)=0. - Vladimir Kruchinin, Jun 08 2011
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1, add(
b(n-j)*irem(j, 2)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(2*n)*(-1)^n:
seq(a(n), n=0..20); # Alois P. Heinz, Feb 11 2023
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MATHEMATICA
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Take[With[{nn=40}, CoefficientList[Series[Cos[Sin[x]], {x, 0, nn}], x] Range[0, nn]!], {1, -1, 2}] (* Harvey P. Dale, Sep 18 2011 *)
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PROG
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(Maxima)
a(n):=sum((2^(2*j+1)*sum((i-n+j)^(2*n)*binomial((2*n-2*j), i)*(-1)^(n-i), i, 0, (n-j))/(2*n-2*j)!), j, 0, n); /* Vladimir Kruchinin, Jun 08 2011 */
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CROSSREFS
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Sequence in context: A318002 A323567 A304865 * A286928 A321042 A244820
Adjacent sequences: A003706 A003707 A003708 * A003710 A003711 A003712
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KEYWORD
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sign
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AUTHOR
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R. H. Hardin, Simon Plouffe
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STATUS
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approved
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