OFFSET
1,4
COMMENTS
"... polynomials like these with nonnegative integral coefficients might reasonably be expected to be generating polynomials for some as yet unknown fatgraph structure."
LINKS
Gheorghe Coserea, Rows n = 1..100, flattened
J. E. Andersen, R. C. Penner, C. M. Reidys, M. S. Waterman, Topological classification and enumeration of RNA structures by genus, J. Math. Biol. 65 (2013) 1261-1278
R. C. Penner, Moduli Spaces and Macromolecules, Bull. Amer. Math. Soc., 53 (2015), 217-268. See p. 259.
FORMULA
The g.f. for column g>0 of triangle A035309 is x^(2*g) * A270790(g) * P_g(x) / (1-4*x)^(3*g-1/2), where P_g(x) is the polynomial associated with row g of the triangle. - Gheorghe Coserea, Apr 17 2016
EXAMPLE
For n = 3 we have P_3(x) = 158*x^2 + 558*x + 135.
For n = 4 we have P_4(x) = 2339*x^3 + 18378*x^2 + 13689*x + 1575.
Triangle begins:
n\k [1] [2] [3] [4] [5] [6]
[1] 1;
[2] 1, 1;
[3] 158 558, 135;
[4] 2339, 18378, 13689, 1575;
[5] 1354, 18908, 28764, 9660, 675;
[6] 617926, 13447818, 34604118, 23001156, 4534875, 218295;
[7] ...
PROG
(PARI)
G = 8; N = 3*G + 1; F = 1; gmax(n) = min(n\2, G);
Q = matrix(N+1, G+1); Qn() = (matsize(Q)[1] - 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, Qn(), 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)));
Kol(g) = vector(Qn()+2-F-2*g, n, polcoeff(Qget(n+F-2 + 2*g, g), F, 'x));
P(g) = {
my(x = 'x + O('x^(G+2)));
return(Pol(Ser(Kol(g)) * (1-4*x)^(3*g-1/2), 'x));
};
concat(vector(G, g, Vec(P(g) / content(P(g))))) \\ Gheorghe Coserea, Apr 17 2016
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Mar 28 2016
EXTENSIONS
More terms from Gheorghe Coserea, Apr 17 2016
STATUS
approved