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A269924
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Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 4.
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13
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225225, 12317877, 12317877, 351683046, 792534015, 351683046, 7034538511, 26225260226, 26225260226, 7034538511, 111159740692, 600398249550, 993494827480, 600398249550, 111159740692, 1480593013900, 10743797911132, 25766235457300, 25766235457300, 10743797911132, 1480593013900, 17302190625720, 160576594766588, 517592962672296, 750260619502310, 517592962672296, 160576594766588, 17302190625720
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OFFSET
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8,1
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COMMENTS
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Row n contains n-7 terms.
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LINKS
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Gheorghe Coserea, Rows n = 8..208, flattened
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.
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EXAMPLE
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Triangle starts:
n\f [1] [2] [3] [4]
[8] 225225;
[9] 12317877, 12317877;
[10] 351683046, 792534015, 351683046;
[11] 7034538511, 26225260226, 26225260226, 7034538511;
[12] ...
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MATHEMATICA
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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}]);
Table[Q[n, f, 4], {n, 8, 14}, {f, 1, n-7}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)
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PROG
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(PARI)
N = 14; G = 4; gmax(n) = min(n\2, G);
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);
concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))
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CROSSREFS
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Columns f=1-10 give: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, A288276 f=6, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10.
Cf. A035309, A269921, A269922, A269923, A269925, A270406, A270407, A270408, A270409, A270410, A270412.
Row sums give A215402 (column 4 of A269919).
Sequence in context: A269116 A252394 A237848 * A288271 A215402 A204743
Adjacent sequences: A269921 A269922 A269923 * A269925 A269926 A269927
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KEYWORD
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nonn,tabl
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AUTHOR
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Gheorghe Coserea, Mar 15 2016
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STATUS
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approved
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