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A269919 Triangle read by rows: T(n,g) is the number of rooted maps with n edges on an orientable surface of genus g. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

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.

FORMULA

(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.

EXAMPLE

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]   ...

MATHEMATICA

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-Fran├žois Alcover, Jul 20 2018 *)

PROG

(PARI)

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))))

CROSSREFS

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.

Sequence in context: A021778 A095178 A289632 * A178418 A249270 A153739

Adjacent sequences:  A269916 A269917 A269918 * A269920 A269921 A269922

KEYWORD

nonn,tabf

AUTHOR

Gheorghe Coserea, Mar 07 2016

STATUS

approved

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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)