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

%I A278120 #32 Nov 24 2016 09:31:16
%S A278120 -1,1,5,25,1033,15745,1599895,12116675,1519810267,5730215335,
%T A278120 2762191322225,155865304045375,275016025098532093,129172331662700995,
%U A278120 358829725148742321475,524011363178245785875,10072731258491333905209253,1181576300858987307102335,68622390512340213600239902775
%N A278120 a(n) is numerator of rational z(n) associated with the non-orientable map asymptotics constant p((n+1)/2).
%H A278120 Gheorghe Coserea, <a href="/A278120/b278120.txt">Table of n, a(n) for n = 0..201</a>
%H A278120 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 A278120 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 A278120 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.
%F A278120 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 A278120 For n=2 we have p(3/2) = 4 * (5/144) * (3/2)^(3/2) / gamma(9/4) = 2/(sqrt(6)*gamma(1/4)).
%e A278120 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)).
%e A278120 n   z(n)                   p((n+1)/2)
%e A278120 0   -1                     3/(sqrt(6)*gamma(3/4))
%e A278120 1   1/12                   1/2
%e A278120 2   5/144                  2/(sqrt(6)*gamma(1/4))
%e A278120 3   25/864                 5/(36*sqrt(Pi))
%e A278120 4   1033/27684             1033/(13860*sqrt(6)*gamma(3/4))
%e A278120 5   15745/248832           3149/442368
%e A278120 6   1599895/11943936       319979/(18796050*sqrt(6)*gamma(1/4))
%e A278120 7   12116675/35831808      484667/(560431872*sqrt(Pi))
%e A278120 8   1519810267/1528823808  1519810267/(4258429005600*sqrt(6)*gamma(3/4))
%e A278120 9   5730215335/1719926784  1146043067/41094783959040
%e A278120 ...
%o A278120 (PARI)
%o A278120 A269418_seq(N) = {
%o A278120   my(y  = vector(N)); y[1] = 1/48;
%o A278120   for (n = 2, N,
%o A278120        y[n] = (25*(n-1)^2-1)/48 * y[n-1] + 1/2*sum(k = 1, n-1, y[k]*y[n-k]));
%o A278120   concat(-1, y);
%o A278120 };
%o A278120 seq(N) = {
%o A278120   my(y = A269418_seq(N), z = vector(N)); z[1] = 1/12;
%o A278120   for (n = 2, N,
%o A278120        my(t1 = if(n%2, 0, y[1+n\2]/3^(n\2)),
%o A278120           t2 = sum(k=1, n-1, z[k]*z[n-k]));
%o A278120       z[n] = (t1 + (5*n-6)/6 * z[n-1] + t2)/2);
%o A278120   concat(-1, z);
%o A278120 };
%o A278120 apply(numerator, seq(18))
%Y A278120 Cf. A269418, A269419, A278121 (denominator).
%K A278120 sign,frac
%O A278120 0,3
%A A278120 _Gheorghe Coserea_, Nov 12 2016