A141353 a(n) = Catalan(n) + 2^n - 0^n.

1, 3, 6, 13, 30, 74, 196, 557, 1686, 5374, 17820, 60834, 212108, 751092, 2690824, 9727613, 35423206, 129775862, 477900844, 1767787478, 6565168996, 24468364172, 91486757944, 343068002258, 1289920924540, 4861979955884

Offset: 0

Author

Paul Barry, Jun 27 2008

Keywords

• easy
• nonn

Comments

Hankel transform is A141354.

Crossrefs

Cf. A000108 (Catalan numbers), A141351.

Programs

Mathematica

f[n_] := Binomial[2n, n]/(n + 1) + 2^n - 0^n; f[0] = 1; Array[f, 29, 0] (* or *)
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 *)

PARI

a(n) = binomial(2*n,n)/(n+1) + 2^n - 0^n; \\ Michel Marcus, Mar 18 2018

Formula

G.f.: c(x)+2x/(1-2x), where c(x) is the g.f. of A000108. [corrected by Paul Barry, Oct 18 2010]

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