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A270407 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus g.

%I A270407 #22 Oct 18 2018 05:11:19
%S A270407 2,22,164,70,1030,1720,5868,24164,6468,31388,256116,258972,160648,
%T A270407 2278660,5554188,1169740,795846,17970784,85421118,66449432,3845020,
%U A270407 129726760,1059255456,1955808460,351683046,18211380,875029804,11270290416,40121261136,26225260226
%N A270407 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 3 faces on an orientable surface of genus g.
%C A270407 Row n contains floor(n/2) terms.
%H A270407 Gheorghe Coserea, <a href="/A270407/b270407.txt">Rows n = 2..102, flattened</a>
%H A270407 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 A270407 Triangle starts:
%e A270407 n\g    [0]          [1]          [2]          [3]          [4]
%e A270407 [2]    2;
%e A270407 [3]    22;
%e A270407 [4]    164,         70;
%e A270407 [5]    1030,        1720;
%e A270407 [6]    5868,        24164,       6468;
%e A270407 [7]    31388,       256116,      258972;
%e A270407 [8]    160648,      2278660,     5554188,     1169740;
%e A270407 [9]    795846,      17970784,    85421118,    66449432;
%e A270407 [10]   3845020,     129726760,   1059255456,  1955808460,  351683046;
%e A270407 [11]   18211380,    875029804,   11270290416, 40121261136, 26225260226;
%e A270407 [12]   ...
%t A270407 Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n < 0 || f < 0 || g < 0 = 0;
%t A270407 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 A270407 T[n_, g_] := Q[n, 3, g];
%t A270407 Table[T[n, g], {n, 2, 11}, {g, 0, Quotient[n, 2]-1}] // Flatten (* _Jean-François Alcover_, Oct 18 2018 *)
%o A270407 (PARI)
%o A270407 N = 11; F = 3; gmax(n) = n\2;
%o A270407 Q = matrix(N + 1, N + 1);
%o A270407 Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
%o A270407 Qset(n, g, v) = { Q[n+1, g+1] = v };
%o A270407 Quadric({x=1}) = {
%o A270407   Qset(0, 0, x);
%o A270407   for (n = 1, length(Q)-1, for (g = 0, gmax(n),
%o A270407     my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
%o A270407        t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
%o A270407        t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
%o A270407        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
%o A270407     Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
%o A270407 };
%o A270407 Quadric('x + O('x^(F+1)));
%o A270407 concat(vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F))))
%Y A270407 Columns k=0-1 give: A000184, A006296.
%K A270407 nonn,tabf
%O A270407 2,1
%A A270407 _Gheorghe Coserea_, Mar 16 2016