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

0,2

Links

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

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

Mathematica

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

Prog

(Magma)
[n le 1 select 1 else 5*Self(n-1) + Catalan(n-1): n in [1..40]]; // 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

Crossrefs

Cf. A000108, A000351.

Sequence in context: A083320 A097139 A034942 * A129171 A082585 A084326

Adjacent sequences: A046711 A046712 A046713 * A046715 A046716 A046717

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved