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