A185010 a(n) = A000108(n)*A015518(n+1), where A000108 are the Catalan numbers and A015518(n) = 2*A015518(n-1) + 3*A015518(n-2).

1, 2, 14, 100, 854, 7644, 72204, 703560, 7037030, 71772844, 743844452, 7810307960, 82909630972, 888316731800, 9593823377880, 104332819539600, 1141523825614470, 12556761952114380, 138785264158902900, 1540516430396559000, 17165754516697206420, 191944345934966132040

Offset: 0

Author

Paul D. Hanna, Dec 26 2012

Keywords

• nonn

Comments

More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |b|>0, |c|>0, S(0)=1, then

Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.

Examples

			G.f.: A(x) = 1 + 1*2*x + 2*7*x^2 + 5*20*x^3 + 14*61*x^4 + 42*182*x^5 + 132*547*x^6 +...+ A000108(n)*A015518(n+1)*x^n +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..925
• S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Crossrefs

Cf. A000108, A001006, A015518.

Programs

Mathematica

CoefficientList[Series[Sqrt[(1 - 4*x - Sqrt[1 - 8*x - 48*x^2])/32]/x, {x, 0, 50}], x] (* G. C. Greubel, Jun 09 2017 *)

PARI

{A000108(n)=binomial(2*n,n)/(n+1)}
{A015518(n)=polcoeff(x/(1-2*x-3*x^2 +x*O(x^n)),n)}
{a(n)=A000108(n)*A015518(n+1)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: sqrt( (1-4*x - sqrt(1-8*x-48*x^2))/32 )/x.

G.f.: sqrt( M(4*x) ), where M(x) is g.f. of A001006. - Werner Schulte, Aug 10 2015

Conjecture: n*(n+1)*a(n) -4*n*(2*n-1)*a(n-1) -12*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Oct 08 2016

G.f: B(m(4z)/4), where B(x) is the g.f. of A000984 and m(x) is the g.f. of A086246. - Alexander Burstein, May 20 2021