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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. 9
-1, 1, 49, 1225, 4412401, 73560025, 245229441961, 7759635184525, 2163099334469560445, 243352176577765537625, 126154825844683612669806743, 307996788703417873806157775, 3816216508144039222348410175181221, 4472139245793702477426700875742975 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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.

EXAMPLE

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

...

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 *)

PROG

(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))

CROSSREFS

Cf. A266240, A269419 (denominator).

Sequence in context: A162204 A162469 A011001 * A115999 A228258 A282930

Adjacent sequences: A269415 A269416 A269417 * A269419 A269420 A269421

KEYWORD

sign,frac

AUTHOR

Gheorghe Coserea, Feb 25 2016

STATUS

approved

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Last modified February 3 20:51 EST 2023. Contains 360045 sequences. (Running on oeis4.)