A270790 Multiplier of polynomial P_n(x) arising from RNA combinatorics.

1, 21, 11, 143, 88179, 111435, 111435, 1361270295, 1137235, 9945637, 16009448637, 19293438101, 3607872924887, 2630885818709841, 195084537038811, 45500599374052095, 1472444896343699846295, 1997334750675075735, 145805436799280528655, 107268833547674677179

Offset: 1

Author

N. J. A. Sloane, Mar 28 2016

Keywords

• nonn
• easy

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..100
• R. C. Penner, Moduli Spaces and Macromolecules, Bull. Amer. Math. Soc., 53 (2015), 217-268. See p. 259.

Crossrefs

Cf. A035309, A035319, A270791.

Programs

PARI

G = 20; 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));
};
vector(G, g, content(P(g)))  \\ Gheorghe Coserea, Apr 17 2016

Formula

a(g) * P_g(0) = A035319(g) = (4*g-1)!!/(2*g+1), where P_g(x) is the polynomial associated with row g of the triangle A270791.

Extensions

More terms from Gheorghe Coserea, Apr 17 2016