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 A269920 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 0. 2
 1, 1, 1, 2, 5, 2, 5, 22, 22, 5, 14, 93, 164, 93, 14, 42, 386, 1030, 1030, 386, 42, 132, 1586, 5868, 8885, 5868, 1586, 132, 429, 6476, 31388, 65954, 65954, 31388, 6476, 429, 1430, 26333, 160648, 442610, 614404, 442610, 160648, 26333, 1430 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Row n contains n+1 terms. LINKS Gheorghe Coserea, Rows n = 0..200, 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]     [7]     [8] [0]    1; [1]    1,      1; [2]    2,      5,      2; [3]    5,      22,     22,     5; [4]    14,     93,     164,    93,     14; [5]    42,     386,    1030,   1030,   386,    42; [6]    132,    1586,   5868,   8885,   5868,   1586,   132; [7]    429,    6476,   31388,  65954,  65954,  31388,  6476,   429; [8]    ... 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, 0], {n, 0, 8}, {f, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *) PROG (PARI) N = 8; G = 0; 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 k=1-6 give: A000108, A000346, A000184, A000365, A000473, A000502. Row sums give A000168 (column 0 of A269919). Cf. A006294 (row maxima). Sequence in context: A089122 A321577 A268789 * A240706 A240642 A240774 Adjacent sequences:  A269917 A269918 A269919 * A269921 A269922 A269923 KEYWORD nonn,tabl AUTHOR Gheorghe Coserea, Mar 14 2016 STATUS approved

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Last modified February 24 10:37 EST 2020. Contains 332209 sequences. (Running on oeis4.)