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A003709 E.g.f. cos(sin(x)) (even powers only).
(Formerly M3986)
5
1, -1, 5, -37, 457, -8169, 188685, -5497741, 197920145, -8541537105, 432381471509, -25340238127989, 1699894200469849, -129076687233903673, 10989863562589199389, -1041327644107761435101, 109095160722852951673633, -12561989444137938396142753 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
|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).
REFERENCES
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).
LINKS
FORMULA
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(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 */
CROSSREFS
Sequence in context: A318002 A323567 A304865 * A361281 A286928 A321042
KEYWORD
sign
AUTHOR
STATUS
approved

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Last modified July 12 22:19 EDT 2024. Contains 374257 sequences. (Running on oeis4.)