A278121 - OEIS

%I #12 Nov 24 2016 09:31:26

%S 1,12,144,864,27648,248832,11943936,35831808,1528823808,1719926784,

%T 220150628352,2972033482752,1141260857376768,106993205379072,

%U 54780521154084864,13695130288521216,42071440246337175552,739537035580145664,6058287395472553279488

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

%H Gheorghe Coserea, <a href="/A278121/b278121.txt">Table of n, a(n) for n = 0..201</a>

%H S. R. Carrell, <a href="https://arxiv.org/abs/1406.1760v2">The Non-Orientable Map Asymptotics Constant pg</a>, arXiv:1406.1760 [math.CO], 2014.

%H Stavros Garoufalidis, Marcos Marino, <a href="https://arxiv.org/abs/0812.1195v4">Universality and asymptotics of graph counting problems in nonorientable surfaces</a>, arXiv:0812.1195 [math.CO], 2008.

%F 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.

%F 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.

%e For n=3 we have p(2) = 4 * (25/864) * (3/2)^2 / gamma(7/2) = 5/(36*sqrt(Pi)).

%e For n=5 we have p(3) = 4 * (15745/248832)*(3/2)^3/gamma(6) = 3149/442368.

%e n   z(n)                   p((n+1)/2)

%e 0   -1                     3/(sqrt(6)*gamma(3/4))

%e 1   1/12                   1/2

%e 2   5/144                  2/(sqrt(6)*gamma(1/4))

%e 3   25/864                 5/(36*sqrt(Pi))

%e 4   1033/27684             1033/(13860*sqrt(6)*gamma(3/4))

%e 5   15745/248832           3149/442368

%e 6   1599895/11943936       319979/(18796050*sqrt(6)*gamma(1/4))

%e 7   12116675/35831808      484667/(560431872*sqrt(Pi))

%e 8   1519810267/1528823808  1519810267/(4258429005600*sqrt(6)*gamma(3/4))

%e 9   5730215335/1719926784  1146043067/41094783959040

%e ...

%o (PARI)

%o A269418_seq(N) = {

%o   my(y  = vector(N)); y[1] = 1/48;

%o   for (n = 2, N,

%o        y[n] = (25*(n-1)^2-1)/48 * y[n-1] + 1/2*sum(k = 1, n-1, y[k]*y[n-k]));

%o   concat(-1, y);

%o };

%o seq(N) = {

%o   my(y = A269418_seq(N), z = vector(N)); z[1] = 1/12;

%o   for (n = 2, N,

%o        my(t1 = if(n%2, 0, y[1+n\2]/3^(n\2)),

%o           t2 = sum(k=1, n-1, z[k]*z[n-k]));

%o       z[n] = (t1 + (5*n-6)/6 * z[n-1] + t2)/2);

%o   concat(-1, z);

%o };

%o apply(denominator, seq(18))

%Y Cf. A269418, A269419, A278120 (numerator).

%K nonn,frac

%O 0,2

%A _Gheorghe Coserea_, Nov 12 2016