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A098075
Threefold convolution of A004148 (the RNA secondary structure numbers) with itself.
1
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
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 13 2004
STATUS
approved