A278120 a(n) is numerator of rational z(n) associated with the non-orientable map asymptotics constant p((n+1)/2).

-1, 1, 5, 25, 1033, 15745, 1599895, 12116675, 1519810267, 5730215335, 2762191322225, 155865304045375, 275016025098532093, 129172331662700995, 358829725148742321475, 524011363178245785875, 10072731258491333905209253, 1181576300858987307102335, 68622390512340213600239902775

Offset

0,3

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..201

S. R. Carrell, The Non-Orientable Map Asymptotics Constant pg, arXiv:1406.1760 [math.CO], 2014.

Stavros Garoufalidis, Marcos Marino, Universality and asymptotics of graph counting problems in nonorientable surfaces, arXiv:0812.1195 [math.CO], 2008.

Formula

a(n) = numerator(z(n)), where z(n) = 1/2 * (y(n/2)/3^(n/2) + (5*n-6)/6 * z(n-1) + Sum {k=1..n-1} z(k)*z(n-k)), with z(0) = -1, y(n) = A269418(n)/A269419(n) and y(n+1/2) = 0 for all n.

p((n+1)/2) = 4 * (A278120(n)/A278121(n)) * (3/2)^((n+1)/2) / gamma((5*n-1)/4), where p((n+1)/2) is the non-orientable map asymptotics constant for type g=(n+1)/2 and gamma is the Gamma function.

Example

For n=2 we have p(3/2) = 4 * (5/144) * (3/2)^(3/2) / gamma(9/4) = 2/(sqrt(6)*gamma(1/4)).
For n=4 we have p(5/2) = 4 * (1033/27648) * (3/2)^(5/2) / gamma(19/4) = 1033/(13860*sqrt(6)*gamma(3/4)).
n   z(n)                   p((n+1)/2)
0   -1                     3/(sqrt(6)*gamma(3/4))
1   1/12                   1/2
2   5/144                  2/(sqrt(6)*gamma(1/4))
3   25/864                 5/(36*sqrt(Pi))
4   1033/27684             1033/(13860*sqrt(6)*gamma(3/4))
5   15745/248832           3149/442368
6   1599895/11943936       319979/(18796050*sqrt(6)*gamma(1/4))
7   12116675/35831808      484667/(560431872*sqrt(Pi))
8   1519810267/1528823808  1519810267/(4258429005600*sqrt(6)*gamma(3/4))
9   5730215335/1719926784  1146043067/41094783959040
...

Prog

(PARI)
A269418_seq(N) = {
  my(y  = vector(N)); y[1] = 1/48;
  for (n = 2, N,
       y[n] = (25*(n-1)^2-1)/48 * y[n-1] + 1/2*sum(k = 1, n-1, y[k]*y[n-k]));
  concat(-1, y);
};
seq(N) = {
  my(y = A269418_seq(N), z = vector(N)); z[1] = 1/12;
  for (n = 2, N,
       my(t1 = if(n%2, 0, y[1+n\2]/3^(n\2)),
          t2 = sum(k=1, n-1, z[k]*z[n-k]));
      z[n] = (t1 + (5*n-6)/6 * z[n-1] + t2)/2);
  concat(-1, z);
};
apply(numerator, seq(18))

Crossrefs

Cf. A269418, A269419, A278121 (denominator).

Sequence in context: A137114 A067212 A061583 * A039780 A328124 A033981

Adjacent sequences: A278117 A278118 A278119 * A278121 A278122 A278123

Keyword

sign,frac

Author

Gheorghe Coserea, Nov 12 2016

Status

approved