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
KEYWORD
nonn,tabl
AUTHOR
Gheorghe Coserea, Mar 14 2016
STATUS
approved