cp's OEIS Frontend

A269419 a(n) is denominator of y(n), where y(n+1) = (25*n^2-1)/48 * y(n) + (1/2)*Sum_{k=1..n}y(k)*y(n+1-k), with y(0) = -1.

%I A269419 #33 Oct 23 2018 10:33:00
%S A269419 1,48,4608,55296,42467328,84934656,21743271936,36691771392,
%T A269419 400771988324352,1352605460594688,16620815899787526144,
%U A269419 779100745302540288,153177439332441840943104,2393397489569403764736,235280546814630667688607744,57441539749665690353664
%N A269419 a(n) is denominator of y(n), where y(n+1) = (25*n^2-1)/48 * y(n) + (1/2)*Sum_{k=1..n}y(k)*y(n+1-k), with y(0) = -1.
%H A269419 Gheorghe Coserea, <a href="/A269419/b269419.txt">Table of n, a(n) for n = 0..500</a>
%H A269419 Edward A. Bender, Zhicheng Gao, L. Bruce Richmond, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r51"> The map asymptotics constant tg</a>, The Electronic Journal of Combinatorics, Volume 15 (2008), Research Paper #R51.
%H A269419 Stavros Garoufalidis, Thang T.Q. Le, Marcos Marino, <a href="http://arxiv.org/abs/0809.2572">Analyticity of the Free Energy of a Closed 3-Manifold</a>, arXiv:0809.2572 [math.GT], 2008.
%F A269419 t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)), where t(g) is the orientable map asymptotics constant and gamma is the Gamma function.
%e A269419 For n=0 we have t(0) = (-1) / (2^(-2)*gamma(-1/2)) = 2/sqrt(Pi).
%e A269419 For n=1 we have t(1) = (1/48) / (2^(-1)*gamma(2))  = 1/24.
%e A269419 n   y(n)                        t(n)
%e A269419 0   -1                          2/sqrt(Pi)
%e A269419 1   1/48                        1/24
%e A269419 2   49/4608                     7/(4320*sqrt(Pi))
%e A269419 3   1225/55296                  245/15925248
%e A269419 4   4412401/42467328            37079/(96074035200*sqrt(Pi))
%e A269419 5   73560025/84934656           38213/14089640214528
%e A269419 6   245229441961/21743271936    5004682489/(92499927372103680000*sqrt(Pi))
%e A269419 7   7759635184525/36691771392   6334396069/20054053184087387013120
%e A269419 ...
%t A269419 y[0] = -1;
%t A269419 y[n_] := y[n] = (25(n-1)^2-1)/48 y[n-1] + 1/2 Sum[y[k] y[n-k], {k, 1, n-1}];
%t A269419 Table[y[n] // Denominator, {n, 0, 15}] (* _Jean-François Alcover_, Oct 23 2018 *)
%o A269419 (PARI)
%o A269419 seq(n) = {
%o A269419   my(y  = vector(n));
%o A269419   y[1] = 1/48;
%o A269419   for (g = 1, n-1,
%o A269419        y[g+1] = (25*g^2-1)/48 * y[g] + 1/2*sum(k = 1, g, y[k]*y[g+1-k]));
%o A269419   return(concat(-1,y));
%o A269419 }
%o A269419 apply(denominator, seq(14))
%Y A269419 Cf. A266240, A269418 (numerator).
%K A269419 nonn,frac
%O A269419 0,2
%A A269419 _Gheorghe Coserea_, Feb 25 2016