A185994 A generalized q-Catalan number for q=2.

1, 2, 6, 26, 150, 1114, 10614, 131002, 2128278, 46294426, 1368718518, 55647242106, 3137452915158, 246751601425242, 27181289502625014, 4205716133932054842, 915890632125187606038, 281117681559174501597466, 121733516122198763782243638

Offset: 0

Author

Paul Barry, Feb 08 2011

Keywords

• nonn
• easy

Comments

Hankel transform is A185995.

Programs

Maple

A185994 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        2*procname(n-1)+add(2^k*procname(k)*procname(n-1-k),k=0..n-2) ;
    end if;
end proc:
seq(A185994(n),n=0..30) ; # R. J. Mathar, Feb 03 2025

Formula

G.f.: 1/(1-2x/(1-x/(1-4x/(1-2x/(1-8x/(1-4x/(1-16x/(1-8x/(1-... (continued fraction).

G.f.: 1/(1-2x-2x^2/(1-5x-8x^2/(1-10x-32x^2/(1-(2^2+2^4)x-2^7x^2/(1-(2^3+2^5)x-2^9x^2/(1-.... (continued fraction).

a(n)=if(n=0,1,2*a(n-1)+sum{k=0..n-2, 2^k*a(k)*a(n-1-k)}).