A269419 -id:A269419 - cp's OEIS frontend

A269919 Triangle read by rows: T(n,g) is the number of rooted maps with n edges on an orientable surface of genus g.

1, 2, 9, 1, 54, 20, 378, 307, 21, 2916, 4280, 966, 24057, 56914, 27954, 1485, 208494, 736568, 650076, 113256, 1876446, 9370183, 13271982, 5008230, 225225, 17399772, 117822512, 248371380, 167808024, 24635754, 165297834, 1469283166, 4366441128

Offset: 0

Gheorghe Coserea, Mar 07 2016

Row n contains floor((n+2)/2) terms.

Equivalently, T(n,g) is the number of rooted bipartite quadrangulations with n faces of an orientable surface of genus g.

			Triangle starts:
n\g    [0]          [1]          [2]          [3]          [4]
[0]    1;
[1]    2;
[2]    9,           1;
[3]    54,          20;
[4]    378,         307,         21;
[5]    2916,        4280,        966;
[6]    24057,       56914,       27954,       1485;
[7]    208494,      736568,      650076,      113256;
[8]    1876446,     9370183,     13271982,    5008230,     225225;
[9]    17399772,    117822512,   248371380,   167808024,   24635754;
[10]   ...

Gheorghe Coserea, Rows n = 0..200, flattened
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Columns g=0-10 give: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
Cf. A269418, A269419.
Same as A238396 except for the zeros.

T[0, 0] = 1; T[n_, g_] /; g<0 || g>n/2 = 0; T[n_, g_] := T[n, g] = ((4n-2)/ 3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n-k)- 1) T[k-1, i] T[n-k-1, g-i], {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
Table[T[n, g], {n, 0, 10}, {g, 0, n/2}] // Flatten (* Jean-François Alcover, Jul 20 2018 *)

N = 9; gmax(n) = n\2;
Q = matrix(N+1, N+1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, N, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric();
concat(vector(N+1, n, vector(1 + gmax(n-1), g, Qget(n-1, g-1))))

(n+1)/6 * T(n, g) = (4*n-2)/3 * T(n-1, g) + (2*n-3)*(2*n-2)*(2*n-1)/12 * T(n-2, g-1) + 1/2 * Sum_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.

For column g, as n goes to infinity we have T(n,g) ~ t(g) * n^(5*(g-1)/2) * 12^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function.

A238396 Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 0, 9, 1, 0, 54, 20, 0, 0, 378, 307, 21, 0, 0, 2916, 4280, 966, 0, 0, 0, 24057, 56914, 27954, 1485, 0, 0, 0, 208494, 736568, 650076, 113256, 0, 0, 0, 0, 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0, 17399772, 117822512, 248371380, 167808024, 24635754, 0, 0, 0, 0, 0, 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0

Offset: 0

Joerg Arndt, Feb 26 2014

			Triangle starts:
00: 1,
01: 2, 0,
02: 9, 1, 0,
03: 54, 20, 0, 0,
04: 378, 307, 21, 0, 0,
05: 2916, 4280, 966, 0, 0, 0,
06: 24057, 56914, 27954, 1485, 0, 0, 0,
07: 208494, 736568, 650076, 113256, 0, 0, 0, 0,
08: 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0,
09: 17399772, 117822512, 248371380, 167808024, 24635754, 0, ...,
10: 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0, ...,
11: 1602117468, 18210135416, 73231116024, 117593590752, 66519597474, 8608033980, 0, ...,
12: 15792300756, 224636864830, 1183803697278, 2675326679856, 2416610807964, 672868675017, 24325703325, 0, ...,
...

David M. Jackson and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.

Joerg Arndt, Table of n, a(n) for n = 0..1325 (rows 0..50, flattened)
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).

Columns k for 0<=k<=10 are: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
Sum of row n is A000698(n+1).
See A267180 for nonorientable analog.
Cf. A269418, A269419.
The triangle without the zeros is A269919.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4n - 2)/3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n - k)-1) T[k-1, i] T[n-k-1, g-i] , {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
Table[T[n, g], {n, 0, 10}, {g, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, after Gheorghe Coserea *)

N=20;
MEM=matrix(N+1,N+1, r,c, -1);  \\ for memoization
Q(n,g)=
{
    if (n<0,  return( (g<=0) ) ); \\ not given in paper
    if (g<0,  return( 0 ) ); \\ not given in paper
    if (n<=0, return( g==0 ) );  \\ as in paper
    my( m = MEM[n+1,g+1] );
    if ( m != -1,  return(m) );  \\ memoized value
    my( t=0 );
    t += (4*n-2)/3 * Q(n-1, g);
    t += (2*n-3)*(2*n-2)*(2*n-1)/12 * Q(n-2, g-1);
    my(l, j);
    t += 1/2*
        sum(k=1, n-1, l=n-k;  \\ l+k == n, both >= 1
            sum(i=0, g, j=g-i;  \\ i+j == g, both >= 0
                (2*k-1)*(2*l-1) * Q(k-1, i) * Q(l-1, j)
            );
        );
    t *= 6/(n+1);
    MEM[n+1, g+1] = t;  \\ memoize
    return(t);
}
for (n=0, N, for (g=0, n, print1(Q(n, g),", "); );  print(); ); /* print triangle */

From Gheorghe Coserea, Mar 11 2016: (Start)
(n+1)/6 * T(n, g) = (4*n-2)/3 * T(n-1, g) + (2*n-3)*(2*n-2)*(2*n-1)/12 * T(n-2, g-1) + 1/2 * Sum_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.
For column g, as n goes to infinity we have T(n,g) ~ t(g) * n^(5*(g-1)/2) * 12^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function.
(End)

A269418 a(n) is numerator of y(n), where y(n+1) = (25n^2-1)/48 y(n) + (1/2)Sum_{k=1..n}y(k)y(n+1-k), with y(0) = -1.

Original entry on oeis.org

-1, 1, 49, 1225, 4412401, 73560025, 245229441961, 7759635184525, 2163099334469560445, 243352176577765537625, 126154825844683612669806743, 307996788703417873806157775, 3816216508144039222348410175181221, 4472139245793702477426700875742975

Offset: 0

Gheorghe Coserea, Feb 25 2016

			For n=0 we have t(0) = (-1) / (2^(-2)*gamma(-1/2)) = 2/sqrt(Pi).
For n=1 we have t(1) = (1/48) / (2^(-1)*gamma(2))  = 1/24.
n   y(n)                        t(n)
0   -1                          2/sqrt(Pi)
1   1/48                        1/24
2   49/4608                     7/(4320*sqrt(Pi))
3   1225/55296                  245/15925248
4   4412401/42467328            37079/(96074035200*sqrt(Pi))
5   73560025/84934656           38213/14089640214528
6   245229441961/21743271936    5004682489/(92499927372103680000*sqrt(Pi))
7   7759635184525/36691771392   6334396069/20054053184087387013120
...

Gheorghe Coserea, Table of n, a(n) for n = 0..187
Edward A. Bender, Zhicheng Gao, L. Bruce Richmond, The map asymptotics constant tg, The Electronic Journal of Combinatorics, Volume 15 (2008), Research Paper #R51.
Stavros Garoufalidis, Thang T.Q. Le, Marcos Marino, Analyticity of the Free Energy of a Closed 3-Manifold, arXiv:0809.2572 [math.GT], 2008.

Cf. A266240, A269419 (denominator).

y[0] = -1;
y[n_] := y[n] = (25(n-1)^2-1)/48 y[n-1] + 1/2 Sum[y[k] y[n-k], {k, 1, n-1}];
Table[y[n] // Numerator, {n, 0, 13}] (* Jean-François Alcover, Oct 23 2018 *)

seq(n) = {
  my(y  = vector(n));
  y[1] = 1/48;
  for (g = 1, n-1,
       y[g+1] = (25*g^2-1)/48 * y[g] + 1/2*sum(k = 1, g, y[k]*y[g+1-k]));
  return(concat(-1,y));
}
apply(numerator, seq(13))

t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)), where t(g) is the orientable map asymptotics constant and gamma is the Gamma function.

A266240 Triangle read by rows: T(n,g) is the number of rooted 2n-face triangulations in an orientable surface of genus g.

Original entry on oeis.org

1, 4, 1, 32, 28, 336, 664, 105, 4096, 14912, 8112, 54912, 326496, 396792, 50050, 786432, 7048192, 15663360, 6722816, 11824384, 150820608, 544475232, 518329776, 56581525, 184549376, 3208396800, 17388675072, 30117189632, 11100235520, 2966845440

Offset: 0

Gheorghe Coserea, Dec 25 2015

Row n contains floor((n+3)/2) terms.

			Triangle starts:
n\g    [0]          [1]          [2]          [3]          [4]
[0]    1;
[1]    4,           1;
[2]    32,          28;
[3]    336,         664,         105;
[4]    4096,        14912,       8112;
[5]    54912,       326496,      396792,      50050;
[6]    786432,      7048192,     15663360,    6722816;
[7]    11824384,    150820608,   544475232,   518329776,   56581525;
[8]    184549376,   3208396800,  17388675072, 30117189632, 11100235520;
[9]    ...

Gheorghe Coserea, Rows n = 0..200, flattened
Edward A. Bender, Zhicheng Gao, L. Bruce Richmond, The map asymptotics constant tg, The Electronic Journal of Combinatorics, Volume 15 (2008), Research Paper #R51.
Zhicheng Gao, A Formula for the Bivariate Map Asymptotics Constants in terms of the Univariate Map Asymptotics Constants, The Electronic Journal of Combinatorics, Volume 17 (2010), Research Paper #R155.
I. P. Goulden and D. M. Jackson, The KP hierarchy, branched covers, and triangulations, Advances in Mathematics, Volume 219, Issue 3, 20 October 2008, Pages 932-951.

Columns k=0-4 give: A002005, A269473, A269474, A269475, A269476.
Row sums give A062980.
Cf. A269418, A269419.

T[n_ /; n >= 0, g_] /; 0 <= g <= (n+1)/2 := f[n, g]/(3n+2); T[, ] = 0; f[n_ /; n >= 1, g_ /; g >= 0] := f[n, g] = 4*(3*n+2)/(n+1)*(n*(3*n-2)*f[n - 2, g-1] + Sum[f[i, h]*f[n-2-i, g-h], {i, -1, n-1}, {h, 0, g}]); f[-1, 0] = 1/2; f[0, 0] = 2; f[, ] = 0; Table[Table[T[n, g], {g, 0, Floor[(n + 1)/2]}], {n, 0, 9}] // Flatten (* Jean-François Alcover, Feb 27 2016 *)

N = 10;
m = matrix(N+2, N+2);
mget(n,g) = {
  if (g < 0 || g > (n+1)/2, return(0));
  return(m[n+2,g+1]);
}
mset(n,g,v) = {
  m[n+2,g+1] = v;
}
Cubic() = {
  mset(-1,0,1/2);
  mset(0,0,2);
  for (n = 1, N,
  for (g = 0, (n+1)\2,
    my(t1 = n * (3*n-2) * mget(n-2, g-1),
       t2 = sum(i = -1, n-1, sum(h = 0, g,
                mget(i,h) * mget(n-2-i, g-h))));
    mset(n, g, 4*(3*n+2)/(n+1) * (t1 + t2))));
  my(a = vector(N+1));
  for (n = 0, N,
    a[n+1] = vector(1 + (n+1)\2);
    for (g = 0, (n+1)\2,
         a[n+1][g+1] = mget(n, g));
    a[n+1] = a[n+1]/(3*n+2));
  return(a);
}
concat(Cubic())

T(n,g) = f(n,g)/(3*n+2) for all n >= 0 and 0 <= g <= (n+1)/2, where f(n,g) satisfies the quadratic recurrence equation f(n,g) = 4*(3*n+2)/(n+1)*(n*(3*n-2)*f(n-2,g-1) + Sum_{i=-1..n-1} Sum_{h=0..g} f(i,h)*f(n-2-i, g-h)) for n >= 1 and g >= 0 with the initial conditions f(-1,0)=1/2, f(0,0)=2 and f(n,g)=0 for g < 0 or g > (n+1)/2.

For column g, as n goes to infinity we have T(n,g) ~ 3*6^((g-1)/2) * t(g) * n^(5*(g-1)/2) * (12*sqrt(3))^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function. - Gheorghe Coserea, Feb 26 2016

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

Original entry on oeis.org

-1, 1, 5, 25, 1033, 15745, 1599895, 12116675, 1519810267, 5730215335, 2762191322225, 155865304045375, 275016025098532093, 129172331662700995, 358829725148742321475, 524011363178245785875, 10072731258491333905209253, 1181576300858987307102335, 68622390512340213600239902775

Offset: 0

Gheorghe Coserea, Nov 12 2016

			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
...

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, A278121 (denominator).

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))

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.

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.

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

Original entry on oeis.org

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.

A278178 a(n) is the numerator of intersection number , n>=2.

Original entry on oeis.org

7, 1225, 1816871, 7723802625, 8591613499103635, 23107999588635836875, 446563431744711553183786875, 17418085137491657842253233328125, 1311214792748795041469921338623972253125, 169160593483166517029276275055903719700625000, 9261817633933021190882924368962406588490587588265625

Offset: 2

Gheorghe Coserea, Nov 13 2016

For 'intersection numbers' see Section 1 in Itzykson and Zuber paper.

			7/240, 1225/144, 1816871/48, 7723802625/8, 8591613499103635/96, ...

Gheorghe Coserea, Table of n, a(n) for n = 2..101
Stavros Garoufalidis, Marcos Marino, Universality and asymptotics of graph counting problems in nonorientable surfaces, arXiv:0812.1195 [math.CO], 2008.
C. Itzykson, J.-B. Zuber, Combinatorics of the Modular Group II: the Kontsevich integrals, arXiv:hep-th/9201001, 1991.

Cf. A269418, A269419, A278179 (denominator).

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+2));
  vector(N, g, (3*g)! * 4^(g+1) / ((5*g)*(5*g+2)) * y[g+2]);
};
apply(numerator, seq(12))

a(n) = numerator((3*n-3)!*4^n/((5*n-5)*(5*n-3)) * A269418(n)/A269419(n)) for n >= 2.

A278179 a(n) is the denominator of intersection number , n>=2.

Original entry on oeis.org

240, 144, 48, 8, 96, 1, 32, 1, 32, 1, 8, 1, 16, 1, 64, 1, 32, 1, 32, 1, 64, 1, 16, 1, 16, 1, 8, 1, 16, 1, 128, 1, 32, 1, 32, 1, 64, 1, 64, 1, 64, 1, 4, 1, 8, 1, 32, 1, 16, 1, 16, 1, 32, 1, 16, 1, 16, 1, 8, 1, 16, 1, 256, 1, 32, 1, 32, 1, 64, 1, 64, 1, 64, 1, 16, 1, 32, 1, 128, 1, 64, 1, 64, 1, 128

Offset: 2

Gheorghe Coserea, Nov 13 2016

For 'intersection numbers' see Section 1 in Itzykson and Zuber paper.

			7/240, 1225/144, 1816871/48, 7723802625/8, 8591613499103635/96, ...

Gheorghe Coserea, Table of n, a(n) for n = 2..1025
Stavros Garoufalidis, Marcos Marino, Universality and asymptotics of graph counting problems in nonorientable surfaces, arXiv:0812.1195 [math.CO], 2008.
C. Itzykson, J.-B. Zuber, Combinatorics of the Modular Group II: the Kontsevich integrals, arXiv:hep-th/9201001, 1991.

Cf. A269418, A269419, A278178 (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+2));
  vector(N, g, (3*g)! * 4^(g+1) / ((5*g)*(5*g+2)) * y[g+2]);
};
apply(denominator, seq(85))

a(n) = denominator((3*n-3)!*4^n/((5*n-5)*(5*n-3)) * A269418(n)/A269419(n)) for n >= 2.

cp's OEIS Frontend

A269919 Triangle read by rows: T(n,g) is the number of rooted maps with n edges on an orientable surface of genus g.

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A238396 Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.

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A269418 a(n) is numerator of y(n), where y(n+1) = (25*n^2-1)/48 * y(n) + (1/2)*Sum_{k=1..n}y(k)*y(n+1-k), with y(0) = -1.

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A266240 Triangle read by rows: T(n,g) is the number of rooted 2n-face triangulations in an orientable surface of genus g.

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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).

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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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A278178 a(n) is the numerator of intersection number , n>=2.

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A278179 a(n) is the denominator of intersection number , n>=2.

A269418 a(n) is numerator of y(n), where y(n+1) = (25n^2-1)/48 y(n) + (1/2)Sum_{k=1..n}y(k)y(n+1-k), with y(0) = -1.