A054687 a(n+1) = 1 + Sum_{k=0..n} binomial(n,k)*a(k)*a(n-k) for n >= 0 with a(0) = 1.

1, 2, 5, 19, 99, 653, 5187, 48117, 510271, 6088301, 80716427, 1177130893, 18727404639, 322769897573, 5990916997611, 119139798166245, 2527255556219167, 56960055683893853, 1359299747696197931, 34240584053654816797, 907911436336049691519, 25277557586594907583733, 737276033151104902965963

Offset: 0

Author

Leroy Quet, Apr 19 2000

Keywords

• easy
• nonn

Examples

			a(5) = 1 + 1*a(0)*a(4) + 4*a(1)*a(3) + 6*a(2)*a(2) + 4*a(3)*a(1) + 1*a(4)*a(0) = 1 + 1*1*99 + 4*2*19 + 6*5*5 + 4*19*2 + 1*99*1 = 653.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..429

Crossrefs

Cf. A052886.

Programs

Mathematica

nmax=20; b = ConstantArray[0,nmax+2]; b[[1]]=1; Do[b[[n+2]] = 1 + Sum[Binomial[n,k]*b[[k+1]]*b[[n-k+1]],{k,0,n}],{n,0,nmax}]; b (* Vaclav Kotesovec, Mar 02 2014 *)

PARI

lista(nn)={my(a=vector(nn)); a[1]=1; for(n=2, nn, a[n]= 1 + sum(k=0, n-2, binomial(n-2, k)*a[k+1]*a[n-k-1])); for(n=1, nn, print1(a[n], ","))}; \\ Petros Hadjicostas, Jun 11 2020

Formula

a(n) ~ n! / r^(n+1), where r = 0.7542714558461742549095127982603266798910769877... is the smallest positive real root of the equation (BesselJ(1, 2) - BesselJ(0, 2))*BesselY(0, 2*exp(r/2)) + BesselJ(0, 2*exp(r/2))*(BesselY(0, 2) - BesselY(1, 2)) = 0. - Vaclav Kotesovec, Mar 02 2014, updated Apr 26 2020

E.g.f. A(x) satisfies A'(x) = exp(x) + A(x)^2 with A(0) = 1. - Petros Hadjicostas, Apr 25 2020

E.g.f.: exp(x/2)*(BesselJ(2, 2)*BesselY(1, 2*exp(x/2)) - BesselJ(1, 2*exp(x/2)) * BesselY(2, 2)) / (BesselJ(2, 2)*BesselY(0, 2*exp(x/2)) - BesselJ(0, 2*exp(x/2)) * BesselY(2, 2)). - Vaclav Kotesovec, Apr 26 2020

Extensions

More terms from James Sellers, Apr 20 2000