A269418 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.

-1, 1, 49, 1225, 4412401, 73560025, 245229441961, 7759635184525, 2163099334469560445, 243352176577765537625, 126154825844683612669806743, 307996788703417873806157775, 3816216508144039222348410175181221, 4472139245793702477426700875742975

Offset: 0

Author

Gheorghe Coserea, Feb 25 2016

Keywords

• sign
• frac

Examples

			For n=0 we have t(0) = (-1) / (2^(-2)*gamma(-1/2)) = 2/sqrt(Pi).
For n=1 we have t(1) = (1/48) / (2^(-1)*gamma(2))  = 1/24.
n   y(n)                        t(n)
0   -1                          2/sqrt(Pi)
1   1/48                        1/24
2   49/4608                     7/(4320*sqrt(Pi))
3   1225/55296                  245/15925248
4   4412401/42467328            37079/(96074035200*sqrt(Pi))
5   73560025/84934656           38213/14089640214528
6   245229441961/21743271936    5004682489/(92499927372103680000*sqrt(Pi))
7   7759635184525/36691771392   6334396069/20054053184087387013120
...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..187
• Edward A. Bender, Zhicheng Gao, L. Bruce Richmond, The map asymptotics constant tg, The Electronic Journal of Combinatorics, Volume 15 (2008), Research Paper #R51.
• Stavros Garoufalidis, Thang T.Q. Le, Marcos Marino, Analyticity of the Free Energy of a Closed 3-Manifold, arXiv:0809.2572 [math.GT], 2008.

Crossrefs

Cf. A266240, A269419 (denominator).

Programs

Mathematica

y[0] = -1;
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}];
Table[y[n] // Numerator, {n, 0, 13}] (* Jean-François Alcover, Oct 23 2018 *)

PARI

seq(n) = {
  my(y  = vector(n));
  y[1] = 1/48;
  for (g = 1, n-1,
       y[g+1] = (25*g^2-1)/48 * y[g] + 1/2*sum(k = 1, g, y[k]*y[g+1-k]));
  return(concat(-1,y));
}
apply(numerator, seq(13))

Formula

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.