|
%I #37 Oct 23 2018 10:06:13
%S -1,1,49,1225,4412401,73560025,245229441961,7759635184525,
%T 2163099334469560445,243352176577765537625,
%U 126154825844683612669806743,307996788703417873806157775,3816216508144039222348410175181221,4472139245793702477426700875742975
%N a(n) is numerator 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 Gheorghe Coserea, <a href="/A269418/b269418.txt">Table of n, a(n) for n = 0..187</a>
%H 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 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 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 For n=0 we have t(0) = (-1) / (2^(-2)*gamma(-1/2)) = 2/sqrt(Pi).
%e For n=1 we have t(1) = (1/48) / (2^(-1)*gamma(2)) = 1/24.
%e n y(n) t(n)
%e 0 -1 2/sqrt(Pi)
%e 1 1/48 1/24
%e 2 49/4608 7/(4320*sqrt(Pi))
%e 3 1225/55296 245/15925248
%e 4 4412401/42467328 37079/(96074035200*sqrt(Pi))
%e 5 73560025/84934656 38213/14089640214528
%e 6 245229441961/21743271936 5004682489/(92499927372103680000*sqrt(Pi))
%e 7 7759635184525/36691771392 6334396069/20054053184087387013120
%e ...
%t y[0] = -1;
%t 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 Table[y[n] // Numerator, {n, 0, 13}] (* _Jean-François Alcover_, Oct 23 2018 *)
%o (PARI)
%o seq(n) = {
%o my(y = vector(n));
%o y[1] = 1/48;
%o for (g = 1, n-1,
%o y[g+1] = (25*g^2-1)/48 * y[g] + 1/2*sum(k = 1, g, y[k]*y[g+1-k]));
%o return(concat(-1,y));
%o }
%o apply(numerator, seq(13))
%Y Cf. A266240, A269419 (denominator).
%K sign,frac
%O 0,3
%A _Gheorghe Coserea_, Feb 25 2016
| 