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Search: id:A001824
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| A001824 |
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Central factorial numbers.
(Formerly M4749 N2031)
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+0
5
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| 1, 10, 259, 12916, 1057221, 128816766, 21878089479, 4940831601000, 1432009163039625, 518142759828635250, 228929627246078500875, 121292816354463333793500, 75908014254880833434338125
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 223, Problem 2.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..50
Index entries for sequences related to factorial numbers
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FORMULA
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E.g.f.: (arcsin x)^3; that is, a_k is the coefficient of x^(2*k+3) in (arcsin x)^3 multiplied by (2*k+3)! and divided by 6 - Joe Keane (jgk(AT)jgk.org)
a(n) = ((2*n+1)!!)^2 * sum[ k=0..n ] (2*k+1)^(-2).
a(n) ~ pi^2*n^2*2^(2*n)*e^(-2*n)*n^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
(-1)^(n-1)*a(n-1) is the coefficient of x^2 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22, 2002.
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EXAMPLE
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(arcsin x)^3 = x^3 + 1/2*x^5 + 37/120*x^7 + 3229/15120*x^9 + ...
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CROSSREFS
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Cf. A002455, A001825, A049033.
Right-hand column 2 in triangle A008956.
Adjacent sequences: A001821 A001822 A001823 this_sequence A001825 A001826 A001827
Sequence in context: A126468 A024293 A120268 this_sequence A024294 A084999 A054593
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Joe Keane (jgk(AT)jgk.org)
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