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%I #14 Oct 18 2018 06:22:07
%S 429,26333,795846,291720,16322085,22764165,259477218,875029804,
%T 205633428,3435601554,22620890127,19678611645,39599553708,
%U 448035881592,925572602058,174437377400,409230997461,7302676928666,29079129795702,19925913354061
%N 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 Row n contains floor((n-5)/2) terms.
%H Gheorghe Coserea, <a href="/A270412/b270412.txt">Rows n = 7..107, 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]
%e [7] 429;
%e [8] 26333;
%e [9] 795846, 291720;
%e [10] 16322085, 22764165;
%e [11] 259477218, 875029804, 205633428;
%e [12] 3435601554, 22620890127, 19678611645;
%e [13] 39599553708, 448035881592, 925572602058, 174437377400;
%e [14] 409230997461, 7302676928666, 29079129795702, 19925913354061;
%e [15] ...
%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, 8, g];
%t Table[T[n, g], {n, 7, 14}, {g, 0, Quotient[n-5, 2]-1}] // Flatten (* _Jean-François Alcover_, Oct 18 2018 *)
%o (PARI)
%o N = 14; F = 8; 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 v = vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F)));
%o concat(v)
%Y Cf. A270411.
%K nonn,tabf
%O 7,1
%A _Gheorghe Coserea_, Mar 17 2016
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