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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.

%I A270410 #17 Oct 18 2018 04:03:15
%S A270410 42,1586,31388,12012,442610,649950,5030004,17970784,4390386,49145460,
%T A270410 344468530,313530000,429166584,5188948072,11270290416,2198596400,
%U A270410 3435601554,65723863196,276221817810,196924458720,25658464260,729734918432,5235847653036,8789123742880,1480593013900
%N 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.
%C A270410 Row n contains floor((n-3)/2) terms.
%H A270410 Gheorghe Coserea, <a href="/A270410/b270410.txt">Rows n = 5..105, flattened</a>
%H A270410 Sean R. Carrell, Guillaume Chapuy, <a href="http://arxiv.org/abs/1402.6300">Simple recurrence formulas to count maps on orientable surfaces</a>, arXiv:1402.6300 [math.CO], 2014.
%e A270410 Triangle starts:
%e A270410 n\g    [0]           [1]           [2]           [3]
%e A270410 [5]    42;
%e A270410 [6]    1586;
%e A270410 [7]    31388,        12012;
%e A270410 [8]    442610,       649950;
%e A270410 [9]    5030004,      17970784,     4390386;
%e A270410 [10]   49145460,     344468530,    313530000;
%e A270410 [11]   429166584,    5188948072,   11270290416,  2198596400;
%e A270410 [12]   3435601554,   65723863196,  276221817810, 196924458720;
%e A270410 [13]   ...
%t A270410 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A270410 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 A270410 T[n_, g_] := Q[n, 6, g];
%t A270410 Table[T[n, g], {n, 5, 13}, {g, 0, Quotient[n-3, 2]-1}] // Flatten (* _Jean-François Alcover_, Oct 18 2018 *)
%o A270410 (PARI)
%o A270410 N = 12; F = 6; gmax(n) = n\2;
%o A270410 Q = matrix(N + 1, N + 1);
%o A270410 Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
%o A270410 Qset(n, g, v) = { Q[n+1, g+1] = v };
%o A270410 Quadric({x=1}) = {
%o A270410   Qset(0, 0, x);
%o A270410   for (n = 1, length(Q)-1, for (g = 0, gmax(n),
%o A270410     my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
%o A270410        t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
%o A270410        t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
%o A270410        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
%o A270410     Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
%o A270410 };
%o A270410 Quadric('x + O('x^(F+1)));
%o A270410 v = vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F)));
%o A270410 concat(v)
%Y A270410 Cf. A270409.
%K A270410 nonn,tabf
%O A270410 5,1
%A A270410 _Gheorghe Coserea_, Mar 17 2016