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Search: id:A005430
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| A005430 |
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Ap\*'ery numbers: n*C(2n,n).
(Formerly M2028)
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+0
15
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| 0, 2, 12, 60, 280, 1260, 5544, 24024, 102960, 437580, 1847560, 7759752, 32449872, 135207800, 561632400, 2326762800, 9617286240, 39671305740, 163352435400, 671560012200, 2756930576400, 11303415363240, 46290177201840
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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sum(n=1,inf,1/a(n))=Pi*sqrt(3)/9 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002
Appears as diagonal in A003506. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2006
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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).
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.25).
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.
I. J. Zucker, On the series $ Sum\sp \infty\sb {k=1}(\sp{2k}\sb {\; k})\sp {-1}k\sp{-n}$ and related sums, J. Number Theory 20 (1985), no. 1, 92-102.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
H. J. H. Tuenter, Walking into an absolute sum
Wadim Zudilin, An elementary proof of Apery's theorem
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MAPLE
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A005430 := n->n*binomial(2*n, n);
with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Composition(n*2+1), size=n)), j=0..n) od: seq(a[n], n=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 03 2007
a:=n->add(binomial(2*n, n), k=1..n): seq(a(n), n=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
a:=n->abs(sum((binomial(-n, n-3)), j=2..n)): seq(a(n), n=2..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
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PROGRAM
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(PARI) a(n)=-(-1)^n*real(polcoeff(serlaplace(x^2*besselh1(1, 2*x)), 2*n)) (from R. Stephan)
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CROSSREFS
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Cf. A002736, A005258, A005259, A005429, A005430. 1/beta(n, n+1) in A061928.
a(n) = A002011(n-1)/2 = 2 * A002457(n-1).
Cf. A001803.
Cf. A003506.
Sequence in context: A009618 A143770 A062478 this_sequence A094434 A001574 A074445
Adjacent sequences: A005427 A005428 A005429 this_sequence A005431 A005432 A005433
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 01 2000
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