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A270408 Triangle read by rows: T(n,g) is the number of rooted maps with n edges and 4 faces on an orientable surface of genus g. 5
5, 93, 1030, 420, 8885, 14065, 65954, 256116, 66066, 442610, 3392843, 3288327, 2762412, 36703824, 85421118, 17454580, 16322085, 344468530, 1558792200, 1171704435, 92400330, 2908358552, 22555934280, 40121261136, 7034538511, 505403910, 22620890127, 276221817810, 945068384880, 600398249550 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

Row n contains floor((n-1)/2) terms.

LINKS

Gheorghe Coserea, Rows n = 3..103, 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\g    [0]          [1]          [2]          [3]          [4]

[3]    5;

[4]    93;

[5]    1030,        420;

[6]    8885,        14065;

[7]    65954,       256116,      66066;

[8]    442610,      3392843,     3288327;

[9]    2762412,     36703824,    85421118,    17454580;

[10]   16322085,    344468530,   1558792200,  1171704435;

[11]   92400330,    2908358552,  22555934280, 40121261136, 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}]);

T[n_, g_] := Q[n, 4, g];

Table[T[n, g], {n, 3, 12}, {g, 0, Quotient[n-1, 2]-1}] // Flatten (* Jean-Fran├žois Alcover, Oct 18 2018 *)

PROG

(PARI)

N = 11; F = 4; gmax(n) = n\2;

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 + O('x^(F+1)));

concat(vector(N+2-F, n, vector(1 + gmax(n-1), g, polcoeff(Qget(n+F-2, g-1), F))))

CROSSREFS

Cf. A000365 (column 0).

Sequence in context: A295407 A152283 A205344 * A000365 A209471 A012784

Adjacent sequences:  A270405 A270406 A270407 * A270409 A270410 A270411

KEYWORD

nonn,tabf

AUTHOR

Gheorghe Coserea, Mar 17 2016

STATUS

approved

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Last modified February 26 08:33 EST 2020. Contains 332277 sequences. (Running on oeis4.)