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%I #30 Jun 13 2017 11:08:31
%S 1,10,10,70,167,70,420,1720,1720,420,2310,14065,24164,14065,2310,
%T 12012,100156,256116,256116,100156,12012,60060,649950,2278660,3392843,
%U 2278660,649950,60060,291720,3944928,17970784,36703824,36703824,17970784
%N Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 1.
%C Row n contains n-1 terms.
%H Gheorghe Coserea, <a href="/A269921/b269921.txt">Rows n = 2..202, 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\f [1] [2] [3] [4] [5] [6] [7]
%e [2] 1;
%e [3] 10, 10;
%e [4] 70, 167, 70;
%e [5] 420, 1720, 1720, 420;
%e [6] 2310, 14065, 24164, 14065, 2310;
%e [7] 12012, 100156, 256116, 256116, 100156, 12012;
%e [8] 60060, 649950, 2278660, 3392843, 2278660, 649950, 60060;
%e [9] ...
%t M = 9; G = 1; gMax[n_] := Min[Quotient[n, 2], G];
%t Q = Array[0&, {M + 1, M + 1}];
%t Qget[n_, g_] := If[g < 0 || g > n/2, 0, Q[[n + 1, g + 1]]];
%t Qset[n_, g_, v_] := (Q[[n + 1, g + 1]] = v );
%t Quadric[x_] := (Qset[0, 0, x]; For[n = 1, n <= Length[Q] - 1, n++, For[g = 0, g <= gMax[n], g++, t1 = (1 + x)*(2*n - 1)/3 * Qget[n - 1, g]; t2 = (2*n - 3)*(2*n - 2)*(2*n - 1)/12 * Qget[n - 2, g - 1]; t3 = 1/2 * Sum[ Sum[(2*k - 1) * (2*(n - k) - 1) * Qget[k - 1, i] * Qget[n - k - 1, g - i], {i, 0, g}], {k, 1, n-1}]; Qset[n, g, (t1 + t2 + t3) * 6/(n+1)]]]);
%t Quadric[x];
%t (List @@@ Table[Qget[n - 1 + 2*G, G] // Expand, {n, 1, M + 1 - 2*G}]) /. x -> 1 // Flatten (* _Jean-François Alcover_, Jun 13 2017, adapted from PARI *)
%o (PARI)
%o N = 9; G = 1; gmax(n) = min(n\2, G);
%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 concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))
%Y Columns f=1-10 give: A002802 f=1, A006295 f=2, A006296 f=3, A288071 f=4, A288072 f=5, A287046 f=6, A287047 f=7, A287048 f=8, A288073 f=9, A288074 f=10.
%Y Row sums give A006300 (column 1 of A269919).
%Y Cf. A006297 (row maxima).
%K nonn,tabl
%O 2,2
%A _Gheorghe Coserea_, Mar 14 2016
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