A278120 -id:A278120 - cp's OEIS frontend

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

Gheorghe Coserea, Nov 12 2016

			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
...

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.

Cf. A269418, A269419, A278120 (numerator).

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))

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.

A316598 a(n) is the number of rooted quadrangulations of the projective plane with n vertices.

Original entry on oeis.org

5, 38, 331, 3098, 30330, 306276, 3163737, 33252050, 354312946, 3817498004, 41510761346, 454882507468, 5017662052868, 55664182358808, 620592559670979, 6949200032479746, 78117065527443654, 881170275583541004, 9970663315885385502, 113137928354523348300

Offset: 1

Gheorghe Coserea, Jul 08 2018

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..301
Stavros Garoufalidis, Marcos Marino, Universality and asymptotics of graph counting problems in nonorientable surfaces, arXiv:0812.1195 [math.CO], 2008-2009.
E. Krasko, A. Omelchenko, Enumeration of r-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps, arXiv:1709.03225 [math.CO], 2017.
E. Krasko, A. Omelchenko, Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus, Discrete Mathematics, Volume 342, Issue 2, February 2019, Pages 584-599.

Cf. A005159, A007137, A278120.

(-6*x + 3*Sqrt[1-12*x] - 2*Sqrt[-36*x + 6*Sqrt[1-12*x] + 3] + 3)/(6*x^2) + O[x]^20 // CoefficientList[#, x]& (* Jean-François Alcover, Feb 24 2019 *)

seq(N) = {
  my(x='x + O('x^(N+2)), c=(1-sqrt(1-12*x))/(6*x));
  Vec((1 - x - 3*x*c - sqrt(1 - 4*x - 4*x*c))/x);
};
seq(20)
\\ test: y='x*Ser(seq(300), 'x); 0 == 3*x^3*y^4 + (12*x^3 - 6*x^2)*y^3 + (18*x^3 + 24*x^2 + x)*y^2 + (12*x^3 + 66*x^2 + 8*x - 2)*y + (3*x^3 + 36*x^2 + 10*x)

G.f. A(x) = (1-x-3*x*c - sqrt(1-4*x-4*x*c))/x, where c=(1-sqrt(1-12*x))/(6*x). (see eqn. (117) in Garoufalidis link)
G.f. y=A(x) satisfies:
0 = 3*x^3*y^4 + 6*x^2*(2*x - 1)*y^3 + x*(18*x^2 + 24*x + 1)*y^2 + 2*(6*x^3 + 33*x^2 + 4*x - 1)*y + x*(3*x^2 + 36*x + 10).
0 = 13*x*(4*x + 1)*(12*x - 1)^3*y''''' + (36864*x^4 + 3840*x^3 + 8832*x^2 + 1556*x - 65)*(12*x - 1)^2*y'''' + 16*(248832*x^4 - 5184*x^3 + 29799*x^2 + 2418*x - 259)*(12*x - 1)*y''' + 72*(1382400*x^4 - 201600*x^3 + 144312*x^2 - 4157*x - 492)*y'' + 144*(276480*x^3 - 51840*x^2 + 31488*x - 979)*y' + 165888*y.
0 = x*(4*x + 1)*(48*x^2 - 6*x + 1)*(12*x - 1)^3*y'''' + 2*(10368*x^4 + 12*x^2 + 47*x - 2)*(12*x - 1)^2*y''' + 6*(86400*x^4 - 10800*x^3 + 2472*x^2 + 132*x - 19)*(12*x - 1)*y'' + (2488320*x^4 - 622080*x^3 + 186192*x^2 - 10728*x - 144)*y' + (10368*x - 648)*y.
a(n) ~ 2^(2*n + 1/2) * 3^(n + 1/2)/ (Gamma(3/4) * n^(5/4)) * (1 - sqrt(3) * Gamma(3/4) / (sqrt(2*Pi) * n^(1/4))). - Vaclav Kotesovec, Oct 04 2019

cp's OEIS Frontend

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

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