A278121 a(n) is denominator of rational z(n) associated with the non-orientable map asymptotics constant p((n+1)/2).

1, 12, 144, 864, 27648, 248832, 11943936, 35831808, 1528823808, 1719926784, 220150628352, 2972033482752, 1141260857376768, 106993205379072, 54780521154084864, 13695130288521216, 42071440246337175552, 739537035580145664, 6058287395472553279488

Offset: 0

Author

Gheorghe Coserea, Nov 12 2016

Keywords

• nonn
• frac

Examples

			For n=3 we have p(2) = 4 * (25/864) * (3/2)^2 / gamma(7/2) = 5/(36*sqrt(Pi)).
For n=5 we have p(3) = 4 * (15745/248832)*(3/2)^3/gamma(6) = 3149/442368.
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
...
		

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.

Crossrefs

Cf. A269418, A269419, A278120 (numerator).

Programs

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(denominator, seq(18))

Formula

a(n) = denominator(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.