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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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

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.

FORMULA

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

EXAMPLE

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]    ...

MATHEMATICA

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-Fran├žois Alcover, Feb 27 2016 *)

PROG

(PARI)

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())

CROSSREFS

Columns k=0-4 give: A002005, A269473, A269474, A269475, A269476.

Row sums give A062980.

Cf. A269418, A269419.

Sequence in context: A303277 A174501 A051142 * A322601 A075804 A059844

Adjacent sequences:  A266237 A266238 A266239 * A266241 A266242 A266243

KEYWORD

nonn,tabf

AUTHOR

Gheorghe Coserea, Dec 25 2015

STATUS

approved

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Last modified July 19 04:25 EDT 2019. Contains 325144 sequences. (Running on oeis4.)