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 A269924 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 4. 13
 225225, 12317877, 12317877, 351683046, 792534015, 351683046, 7034538511, 26225260226, 26225260226, 7034538511, 111159740692, 600398249550, 993494827480, 600398249550, 111159740692, 1480593013900, 10743797911132, 25766235457300, 25766235457300, 10743797911132, 1480593013900, 17302190625720, 160576594766588, 517592962672296, 750260619502310, 517592962672296, 160576594766588, 17302190625720 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 8,1 COMMENTS Row n contains n-7 terms. LINKS Gheorghe Coserea, Rows n = 8..208, 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] [8]  225225; [9]  12317877,     12317877; [10] 351683046,    792534015,    351683046; [11] 7034538511,   26225260226,  26225260226,  7034538511; [12] ... 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, 4], {n, 8, 14}, {f, 1, n-7}] // Flatten (* Jean-François Alcover, Aug 10 2018 *) PROG (PARI) N = 14; G = 4; 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: A288271 f=1, A288272 f=2, A288273 f=3, A288274 f=4, A288275 f=5, A288276 f=6, A288277 f=7, A288278 f=8, A288279 f=9, A288280 f=10. Cf. A035309, A269921, A269922, A269923, A269925, A270406, A270407, A270408, A270409, A270410, A270412. Row sums give A215402 (column 4 of A269919). Sequence in context: A269116 A252394 A237848 * A288271 A215402 A204743 Adjacent sequences:  A269921 A269922 A269923 * A269925 A269926 A269927 KEYWORD nonn,tabl AUTHOR Gheorghe Coserea, Mar 15 2016 STATUS approved

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Last modified May 25 17:48 EDT 2022. Contains 354071 sequences. (Running on oeis4.)