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A269921 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. 15
1, 10, 10, 70, 167, 70, 420, 1720, 1720, 420, 2310, 14065, 24164, 14065, 2310, 12012, 100156, 256116, 256116, 100156, 12012, 60060, 649950, 2278660, 3392843, 2278660, 649950, 60060, 291720, 3944928, 17970784, 36703824, 36703824, 17970784 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Row n contains n-1 terms.

LINKS

Gheorghe Coserea, Rows n = 2..202, 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]

[2]    1;

[3]    10,      10;

[4]    70,      167,     70;

[5]    420,     1720,    1720,    420;

[6]    2310,    14065,   24164,   14065,   2310;

[7]    12012,   100156,  256116,  256116,  100156,  12012;

[8]    60060,   649950,  2278660, 3392843, 2278660, 649950,  60060;

[9]    ...

MATHEMATICA

M = 9; G = 1; gMax[n_] := Min[Quotient[n, 2], G];

Q = Array[0&, {M + 1, M + 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_] := (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)]]]);

Quadric[x];

(List @@@ Table[Qget[n - 1 + 2*G, G] // Expand, {n, 1, M + 1 - 2*G}]) /. x -> 1 // Flatten (* Jean-Fran├žois Alcover, Jun 13 2017, adapted from PARI *)

PROG

(PARI)

N = 9; G = 1; 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: 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.

Row sums give A006300 (column 1 of A269919).

Cf. A006297 (row maxima).

Sequence in context: A056473 A241869 A243126 * A219797 A255744 A165831

Adjacent sequences:  A269918 A269919 A269920 * A269922 A269923 A269924

KEYWORD

nonn,tabl

AUTHOR

Gheorghe Coserea, Mar 14 2016

STATUS

approved

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Last modified January 21 10:26 EST 2020. Contains 331105 sequences. (Running on oeis4.)