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

Author

Emeric Deutsch, Sep 13 2004

Keywords

• nonn

Links

• 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.

Crossrefs

Cf. A004148.

Programs

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);

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