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A270410 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 6 faces on an orientable surface of genus g. 6
42, 1586, 31388, 12012, 442610, 649950, 5030004, 17970784, 4390386, 49145460, 344468530, 313530000, 429166584, 5188948072, 11270290416, 2198596400, 3435601554, 65723863196, 276221817810, 196924458720, 25658464260, 729734918432, 5235847653036, 8789123742880, 1480593013900 (list; graph; refs; listen; history; text; internal format)
OFFSET

5,1

COMMENTS

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

LINKS

Gheorghe Coserea, Rows n = 5..105, flattened

Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

EXAMPLE

Triangle starts:

n\g    [0]           [1]           [2]           [3]

[5]    42;

[6]    1586;

[7]    31388,        12012;

[8]    442610,       649950;

[9]    5030004,      17970784,     4390386;

[10]   49145460,     344468530,    313530000;

[11]   429166584,    5188948072,   11270290416,  2198596400;

[12]   3435601554,   65723863196,  276221817810, 196924458720;

[13]   ...

MATHEMATICA

Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;

Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1)((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3)(2n-2)(2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1)(2l-1) Q[k-1, u, i] Q[l-1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);

T[n_, g_] := Q[n, 6, g];

Table[T[n, g], {n, 5, 13}, {g, 0, Quotient[n-3, 2]-1}] // Flatten (* Jean-Fran├žois Alcover, Oct 18 2018 *)

PROG

(PARI)

N = 12; F = 6; 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, length(Q)-1, 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('x + O('x^(F+1)));

v = vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F)));

concat(v)

CROSSREFS

Cf. A270409.

Sequence in context: A258492 A067638 A155021 * A000502 A215545 A004997

Adjacent sequences:  A270407 A270408 A270409 * A270411 A270412 A270413

KEYWORD

nonn,tabf

AUTHOR

Gheorghe Coserea, Mar 17 2016

STATUS

approved

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Last modified January 23 19:45 EST 2020. Contains 331175 sequences. (Running on oeis4.)