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A270790 Multiplier of polynomial P_n(x) arising from RNA combinatorics. 3
1, 21, 11, 143, 88179, 111435, 111435, 1361270295, 1137235, 9945637, 16009448637, 19293438101, 3607872924887, 2630885818709841, 195084537038811, 45500599374052095, 1472444896343699846295, 1997334750675075735, 145805436799280528655, 107268833547674677179 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 19 11:26 EDT 2019. Contains 328216 sequences. (Running on oeis4.)