Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #15 Mar 19 2018 05:41:36
%S 1,3,6,13,30,74,196,557,1686,5374,17820,60834,212108,751092,2690824,
%T 9727613,35423206,129775862,477900844,1767787478,6565168996,
%U 24468364172,91486757944,343068002258,1289920924540,4861979955884
%N a(n) = Catalan(n) + 2^n - 0^n.
%C Hankel transform is A141354.
%F G.f.: c(x)+2x/(1-2x), where c(x) is the g.f. of A000108. [corrected by _Paul Barry_, Oct 18 2010]
%F Conjecture: (n+1)*a(n) + 2*(-4*n+1)*a(n-1) + 4*(5*n-7)*a(n-2) + 8*(-2*n+5)*a(n-3) = 0. - _R. J. Mathar_, Nov 15 2012
%t f[n_] := Binomial[2n, n]/(n + 1) + 2^n - 0^n; f[0] = 1; Array[f, 29, 0] (* or *)
%t CoefficientList[ Series[1 + 1/2 (-4 + 2/(1 - 2x) + (1 - Sqrt[1 - 4x])/x), {x, 0, 28}], x] (* _Robert G. Wilson v_, Mar 18 2018 *)
%o (PARI) a(n) = binomial(2*n,n)/(n+1) + 2^n - 0^n; \\ _Michel Marcus_, Mar 18 2018
%Y Cf. A000108 (Catalan numbers), A141351.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jun 27 2008