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

1,2

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.

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.

Prog

(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

Crossrefs

Cf. A035309, A035319, A270791.

Sequence in context: A086981 A114011 A300943 * A300887 A301497 A258083

Adjacent sequences: A270787 A270788 A270789 * A270791 A270792 A270793

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 28 2016

Extensions

More terms from Gheorghe Coserea, Apr 17 2016

Status

approved