A046714 Convolution of A000108 (Catalan) with A000351 (powers of 5).

1, 6, 32, 165, 839, 4237, 21317, 107014, 536500, 2687362, 13453606, 67326816, 336842092, 1684953360, 8427441240, 42146901045, 210769862895, 1053978959265, 5270372435025, 26353629438315, 131774711311995

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A000351.

Programs

Magma

[n le 1 select 1 else 5*Self(n-1) + Catalan(n-1): n in [1..40]]; // G. C. Greubel, Jul 28 2024

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x*(1-5*x)), {x,0,40}], x] (* G. C. Greubel, Jul 28 2024 *)

SageMath

@CachedFunction
def A046714(n): return 1 if n==1 else 5*A046714(n-1) + catalan_number(n-1)
[A046714(n) for n in range(1,41)] # G. C. Greubel, Jul 28 2024

Formula

a(n) = Sum_{k=0..n} A000108(k)*5^(n-k).

a(n) = 5*a(n-1) + C(n), a(0) = 1.

G.f.: c(x)/(1-5*x), where c(x) = g.f. for Catalan numbers A000108.

Homogeneous recursion: a(n) = (3*(3*n+1)/(n+1))*a(n-1) - (10*(2*n-1)/(n+1))*a(n-2), a(-1) := 0, a(0)=1, n >= 1.

Hypergeometric 2F1 form: 2*a(n) = 5^(n+1) - binomial(2*(n+1), n+1) * hypergeom([ -n-1, 1 ], [ 1/2 ], -1/4).

a(n) ~ (5-sqrt(5))/2 * 5^n. - Vaclav Kotesovec, Jul 07 2016