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A002736
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Apéry numbers: n^2*C(2n,n).
(Formerly M2136 N0848)
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12
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0, 2, 24, 180, 1120, 6300, 33264, 168168, 823680, 3938220, 18475600, 85357272, 389398464, 1757701400, 7862853600, 34901442000, 153876579840, 674412197580, 2940343837200, 12759640231800, 55138611528000, 237371722628040, 1018383898440480
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| sum(n>=1, 1/a(n) ) = Pi^2/18 (Euler) - Benoit Cloitre, Apr 07 2002
Let H be the n-by-n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-1 of B equals -a(n-1). - T. D. Noe, May 01 2011
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REFERENCES
| J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. J. van der Poorten, A proof that Euler missed...Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/1979), 195-203.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
H. J. H. Tuenter, Walking into an absolute sum
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FORMULA
| G.f.: x*(4*x+2)/((1-4*x)^(5/2)) [From Marco A. Cisneros Guevara, July 25 2011]
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MAPLE
| with(combinat):for n from 0 to 22 do printf(`%d, `, n*sum(binomial(2*n, n), k=1..n)) od: - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007
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MATHEMATICA
| CoefficientList[ Series[x (4 x + 2)/(1 - 4 x)^(5/2), {x, 0, 20}], x] (* Robert G. Wilson v, Aug 8 2011 *)
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PROG
| (Mupad) combinat::catalan(n)*(n+1)*n^2 $ n = 0..36 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 17 2007
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CROSSREFS
| Cf. A002736, A005258, A005259, A005429, A005430.
Sequence in context: A157053 A052411 A073066 * A131972 A059387 A126190
Adjacent sequences: A002733 A002734 A002735 * A002737 A002738 A002739
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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