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A270412 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus g.

%I A270412 #14 Oct 18 2018 06:22:07
%S A270412 429,26333,795846,291720,16322085,22764165,259477218,875029804,
%T A270412 205633428,3435601554,22620890127,19678611645,39599553708,
%U A270412 448035881592,925572602058,174437377400,409230997461,7302676928666,29079129795702,19925913354061
%N A270412 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 8 faces on an orientable surface of genus g.
%C A270412 Row n contains floor((n-5)/2) terms.
%H A270412 Gheorghe Coserea, <a href="/A270412/b270412.txt">Rows n = 7..107, flattened</a>
%H A270412 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 A270412 Triangle starts:
%e A270412 n\g    [0]              [1]              [2]              [3]
%e A270412 [7]    429;
%e A270412 [8]    26333;
%e A270412 [9]    795846,          291720;
%e A270412 [10]   16322085,        22764165;
%e A270412 [11]   259477218,       875029804,       205633428;
%e A270412 [12]   3435601554,      22620890127,     19678611645;
%e A270412 [13]   39599553708,     448035881592,    925572602058,    174437377400;
%e A270412 [14]   409230997461,    7302676928666,   29079129795702,  19925913354061;
%e A270412 [15]   ...
%t A270412 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A270412 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 A270412 T[n_, g_] := Q[n, 8, g];
%t A270412 Table[T[n, g], {n, 7, 14}, {g, 0, Quotient[n-5, 2]-1}] // Flatten (* _Jean-François Alcover_, Oct 18 2018 *)
%o A270412 (PARI)
%o A270412 N = 14; F = 8; gmax(n) = n\2;
%o A270412 Q = matrix(N + 1, N + 1);
%o A270412 Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
%o A270412 Qset(n, g, v) = { Q[n+1, g+1] = v };
%o A270412 Quadric({x=1}) = {
%o A270412   Qset(0, 0, x);
%o A270412   for (n = 1, length(Q)-1, for (g = 0, gmax(n),
%o A270412     my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
%o A270412        t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
%o A270412        t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
%o A270412        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
%o A270412     Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
%o A270412 };
%o A270412 Quadric('x + O('x^(F+1)));
%o A270412 v = vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F)));
%o A270412 concat(v)
%Y A270412 Cf. A270411.
%K A270412 nonn,tabf
%O A270412 7,1
%A A270412 _Gheorghe Coserea_, Mar 17 2016