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 A269922 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 2. 14
 21, 483, 483, 6468, 15018, 6468, 66066, 258972, 258972, 66066, 570570, 3288327, 5554188, 3288327, 570570, 4390386, 34374186, 85421118, 85421118, 34374186, 4390386, 31039008, 313530000, 1059255456, 1558792200, 1059255456, 313530000, 31039008 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 4,1 COMMENTS Row n contains n-3 terms. LINKS Gheorghe Coserea, Rows n = 4..204, flattened Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014. EXAMPLE Triangle starts: n\f  [1]        [2]        [3]        [4]        [5]        [6] [4]  21; [5]  483,       483; [6]  6468,      15018,     6468; [7]  66066,     258972,    258972,    66066; [8]  570570,    3288327,   5554188,   3288327,   570570; [9]  4390386,   34374186,  85421118,  85421118,  34374186,  4390386; [10] ... MATHEMATICA Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0; 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}]); Table[Q[n, f, 2], {n, 4, 10}, {f, 1, n-3}] // Flatten (* Jean-François Alcover, Aug 10 2018 *) PROG (PARI) N = 10; G = 2; gmax(n) = min(n\2, G); Q = matrix(N + 1, N + 1); Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) }; Qset(n, g, v) = { Q[n+1, g+1] = v }; Quadric({x=1}) = {   Qset(0, 0, x);   for (n = 1, length(Q)-1, for (g = 0, gmax(n),     my(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(k = 1, n-1, sum(i = 0, g,        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));     Qset(n, g, (t1 + t2 + t3) * 6/(n+1)))); }; Quadric('x); concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G)))) CROSSREFS Columns f=1-10 give: A006298 f=1, A288082 f=2, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10. Row sums give A006301 (column 2 of A269919). Cf. A006299 (row maxima), A269921. Sequence in context: A307600 A025603 A296586 * A006298 A089907 A015695 Adjacent sequences:  A269919 A269920 A269921 * A269923 A269924 A269925 KEYWORD nonn,tabl AUTHOR Gheorghe Coserea, Mar 15 2016 STATUS approved

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Last modified January 29 08:04 EST 2020. Contains 331337 sequences. (Running on oeis4.)