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A270411 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus g.

%I A270411 #14 Oct 18 2018 05:11:28
%S A270411 132,6476,160648,60060,2762412,3944928,37460376,129726760,31039008,
%T A270411 429166584,2908358552,2583699888,4331674512,50534154408,106853266632,
%U A270411 20465052608,39599553708,729734918432,2979641557620,2079913241120
%N A270411 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 7 faces on an orientable surface of genus g.
%C A270411 Row n contains floor((n-4)/2) terms.
%H A270411 Gheorghe Coserea, <a href="/A270411/b270411.txt">Rows n = 6..106, flattened</a>
%H A270411 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 A270411 Triangle starts:
%e A270411 n\g    [0]              [1]              [2]              [3]
%e A270411 [6]    132;
%e A270411 [7]    6476;
%e A270411 [8]    160648,          60060;
%e A270411 [9]    2762412,         3944928;
%e A270411 [10]   37460376,        129726760,       31039008;
%e A270411 [11]   429166584,       2908358552,      2583699888;
%e A270411 [12]   4331674512,      50534154408,     106853266632,    20465052608;
%e A270411 [13]   39599553708,     729734918432,    2979641557620,   2079913241120;
%e A270411 [14]   ...
%t A270411 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A270411 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 A270411 T[n_, g_] := Q[n, 7, g];
%t A270411 Table[T[n, g], {n, 6, 13}, {g, 0, Quotient[n-4, 2]-1}] // Flatten
%o A270411 (PARI)
%o A270411 N = 13; F = 7; gmax(n) = n\2;
%o A270411 Q = matrix(N + 1, N + 1);
%o A270411 Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
%o A270411 Qset(n, g, v) = { Q[n+1, g+1] = v };
%o A270411 Quadric({x=1}) = {
%o A270411   Qset(0, 0, x);
%o A270411   for (n = 1, length(Q)-1, for (g = 0, gmax(n),
%o A270411     my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
%o A270411        t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
%o A270411        t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
%o A270411        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
%o A270411     Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
%o A270411 };
%o A270411 Quadric('x + O('x^(F+1)));
%o A270411 v = vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F)));
%o A270411 concat(v)
%Y A270411 Cf. A270410.
%K A270411 nonn,tabf
%O A270411 6,1
%A A270411 _Gheorghe Coserea_, Mar 17 2016