cp's OEIS Frontend

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

%I A098075 #15 Jul 24 2022 11:53:58
%S A098075 1,3,6,13,30,69,160,375,885,2102,5022,12060,29095,70485,171399,418220,
%T A098075 1023663,2512761,6184253,15257262,37725972,93477778,232069116,
%U A098075 577179078,1437926977,3587977293,8966170056,22437282917,56221762626,141051397725
%N A098075 Threefold convolution of A004148 (the RNA secondary structure numbers) with itself.
%H A098075 I. L. Hofacker, P. Schuster and P. F. Stadler, <a href="https://doi.org/10.1016/S0166-218X(98)00073-0">Combinatorics of RNA secondary structures</a>, Discrete Appl. Math., 88, 1998, 207-237.
%H A098075 P. R. Stein and M. S. Waterman, <a href="https://doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1979), 261-272.
%H A098075 M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.mat.univie.ac.at/~slc/opapers/s08viennot.html">Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire</a>, Sem. Loth. Comb. B08l (1984) 79-86.
%F A098075 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.
%F A098075 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.
%F A098075 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
%F A098075 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
%p A098075 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);
%Y A098075 Cf. A004148.
%K A098075 nonn
%O A098075 0,2
%A A098075 _Emeric Deutsch_, Sep 13 2004