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
Examples
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 ...
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.
Programs
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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))