A098075 Threefold convolution of A004148 (the RNA secondary structure numbers) with itself.

1, 3, 6, 13, 30, 69, 160, 375, 885, 2102, 5022, 12060, 29095, 70485, 171399, 418220, 1023663, 2512761, 6184253, 15257262, 37725972, 93477778, 232069116, 577179078, 1437926977, 3587977293, 8966170056, 22437282917, 56221762626, 141051397725

Offset

0,2

Links

Table of n, a(n) for n=0..29.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86.

Formula

a(n) = 3*Sum_{k=ceiling((n+1)/2)..n} binomial(k, n-k)*binomial(k+2, 3+n-k)/k, n >= 1, a(0)=1.

G.f.: f^3, where f = (1 - z + z^2 - sqrt(1 - 2*z - z^2 - 2*z^3 + z^4))/(2z^2) is the g.f. of A004148.

a(n) ~ 3 * 5^(1/4) * phi^(2*n+6) / (2 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2022

D-finite with recurrence n^2*(n+6)*a(n) -n*(2*n+5)*(n+2)*a(n-1) -(n+1)*(n^2+2*n-16)*a(n-2) -n*(n+2)*(2*n-1)*a(n-3) +(n-4)*(n+2)^2*a(n-4)=0. - R. J. Mathar, Jul 24 2022

Maple

a:=proc(n) if n=0 then 1 else 3*sum(binomial(k, n-k)*binomial(k+2, 3+n-k)/k, k=ceil((n+1)/2)..n) fi end: seq(a(n), n=0..30);

Crossrefs

Cf. A004148.

Sequence in context: A108639 A327795 A087218 * A137584 A201631 A125267

Adjacent sequences: A098072 A098073 A098074 * A098076 A098077 A098078

Keyword

nonn

Author

Emeric Deutsch, Sep 13 2004

Status

approved