cp's OEIS Frontend

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

%I A185994 #10 Feb 03 2025 09:43:39
%S A185994 1,2,6,26,150,1114,10614,131002,2128278,46294426,1368718518,
%T A185994 55647242106,3137452915158,246751601425242,27181289502625014,
%U A185994 4205716133932054842,915890632125187606038,281117681559174501597466,121733516122198763782243638
%N A185994 A generalized q-Catalan number for q=2.
%C A185994 Hankel transform is A185995.
%F A185994 G.f.: 1/(1-2x/(1-x/(1-4x/(1-2x/(1-8x/(1-4x/(1-16x/(1-8x/(1-... (continued fraction).
%F A185994 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).
%F A185994 a(n)=if(n=0,1,2*a(n-1)+sum{k=0..n-2, 2^k*a(k)*a(n-1-k)}).
%p A185994 A185994 := proc(n)
%p A185994     option remember ;
%p A185994     if n = 0 then
%p A185994         1;
%p A185994     else
%p A185994         2*procname(n-1)+add(2^k*procname(k)*procname(n-1-k),k=0..n-2) ;
%p A185994     end if;
%p A185994 end proc:
%p A185994 seq(A185994(n),n=0..30) ; # _R. J. Mathar_, Feb 03 2025
%K A185994 nonn,easy
%O A185994 0,2
%A A185994 _Paul Barry_, Feb 08 2011