A270791 Triangle read by rows: coefficients of polynomials P_n(x) arising from RNA combinatorics.

1, 1, 1, 158, 558, 135, 2339, 18378, 13689, 1575, 1354, 18908, 28764, 9660, 675, 617926, 13447818, 34604118, 23001156, 4534875, 218295, 525206428, 16383145284, 63886133214, 70424606988, 26926791930, 3567422250, 127702575, 50531787, 2134308548, 11735772822, 19350632598, 12106771137, 3063221550, 295973325, 8292375

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

Cf. A035309, A035319, A270790.

Sequence in context: A072555 A056088 A189813 * A180096 A250996 A280483

Adjacent sequences: A270788 A270789 A270790 * A270792 A270793 A270794

Keyword

nonn,easy,tabl

Author

N. J. A. Sloane, Mar 28 2016

Extensions

More terms from Gheorghe Coserea, Apr 17 2016

Status

approved