%I A005430 M2028
%S A005430 0,2,12,60,280,1260,5544,24024,102960,437580,1847560,7759752,32449872,
%T A005430 135207800,561632400,2326762800,9617286240,39671305740,163352435400,
%U A005430 671560012200,2756930576400,11303415363240,46290177201840
%N A005430 Ap\*'ery numbers: n*C(2n,n).
%C A005430 sum(n=1,inf,1/a(n))=Pi*sqrt(3)/9 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 07 2002
%C A005430 Appears as diagonal in A003506. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 12 2006
%D A005430 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005430 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 78, (3.5.25).
%D A005430 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.
%D A005430 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.
%H A005430 T. D. Noe, <a href="b005430.txt">Table of n, a(n) for n=0..200</a>
%H A005430 H. J. H. Tuenter, <a href="http://arXiv.org/abs/math.NT/0606080">Walking 
               into an absolute sum</a>
%H A005430 Wadim Zudilin, <a href="http://arXiv.org/abs/math/0202159">An elementary 
               proof of Apery's theorem</a>
%p A005430 A005430 := n->n*binomial(2*n,n);
%p A005430 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
%p A005430 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
%p A005430 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
%o A005430 (PARI) a(n)=-(-1)^n*real(polcoeff(serlaplace(x^2*besselh1(1,2*x)),2*n)) 
               (from R. Stephan)
%Y A005430 Cf. A002736, A005258, A005259, A005429, A005430. 1/beta(n, n+1) in A061928.
%Y A005430 a(n) = A002011(n-1)/2 = 2 * A002457(n-1).
%Y A005430 Cf. A001803.
%Y A005430 Cf. A003506.
%Y A005430 Sequence in context: A009618 A143770 A062478 this_sequence A094434 A001574 
               A074445
%Y A005430 Adjacent sequences: A005427 A005428 A005429 this_sequence A005431 A005432 
               A005433
%K A005430 nonn,easy,nice
%O A005430 0,2
%A A005430 Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A005430 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 01 2000