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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.
18
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
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-François 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: A241869 A243126 A377189 * A219797 A377215 A255744
KEYWORD
nonn,tabl
AUTHOR
Gheorghe Coserea, Mar 14 2016
STATUS
approved