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%I #34 Jul 20 2018 03:44:59
%S 1,2,9,1,54,20,378,307,21,2916,4280,966,24057,56914,27954,1485,208494,
%T 736568,650076,113256,1876446,9370183,13271982,5008230,225225,
%U 17399772,117822512,248371380,167808024,24635754,165297834,1469283166,4366441128
%N Triangle read by rows: T(n,g) is the number of rooted maps with n edges on an orientable surface of genus g.
%C Row n contains floor((n+2)/2) terms.
%C Equivalently, T(n,g) is the number of rooted bipartite quadrangulations with n faces of an orientable surface of genus g.
%H Gheorghe Coserea, <a href="/A269919/b269919.txt">Rows n = 0..200, 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.
%F (n+1)/6 * T(n, g) = (4*n-2)/3 * T(n-1, g) + (2*n-3)*(2*n-2)*(2*n-1)/12 * T(n-2, g-1) + 1/2 * Sum_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.
%F For column g, as n goes to infinity we have T(n,g) ~ t(g) * n^(5*(g-1)/2) * 12^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function.
%e Triangle starts:
%e n\g [0] [1] [2] [3] [4]
%e [0] 1;
%e [1] 2;
%e [2] 9, 1;
%e [3] 54, 20;
%e [4] 378, 307, 21;
%e [5] 2916, 4280, 966;
%e [6] 24057, 56914, 27954, 1485;
%e [7] 208494, 736568, 650076, 113256;
%e [8] 1876446, 9370183, 13271982, 5008230, 225225;
%e [9] 17399772, 117822512, 248371380, 167808024, 24635754;
%e [10] ...
%t T[0, 0] = 1; T[n_, g_] /; g<0 || g>n/2 = 0; T[n_, g_] := T[n, g] = ((4n-2)/ 3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n-k)- 1) T[k-1, i] T[n-k-1, g-i], {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
%t Table[T[n, g], {n, 0, 10}, {g, 0, n/2}] // Flatten (* _Jean-François Alcover_, Jul 20 2018 *)
%o (PARI)
%o N = 9; 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, N, 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();
%o concat(vector(N+1, n, vector(1 + gmax(n-1), g, Qget(n-1, g-1))))
%Y Columns g=0-10 give: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
%Y Cf. A269418, A269419.
%Y Same as A238396 except for the zeros.
%K nonn,tabf
%O 0,2
%A _Gheorghe Coserea_, Mar 07 2016
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