A046814 Row sums of triangle A046527.

1, 2, 8, 37, 179, 881, 4369, 21746, 108444, 541362, 2704158, 13512392, 67534828, 337584992, 1687627800, 8437136085, 42182258715, 210899507685, 1054456597965, 5272139698215, 26360193558735, 131799177579015

Offset

0,2

Links

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

Formula

G.f.: c(x) * (1-4*x) / (1-5*x), where c(x) = g.f. for Catalan A000108.

a(n) = C(n) + A046714(n-1) with A046714(-1) = 0 and C(n) = A000108(n) are the Catalan numbers.

a(n) = C(n) + (5^n - A046748(n))/2.

a(n) = 5*a(n-1) - 3*C(n)/(2*n-1), a(0)=1.

D-finite with recurrence a(n) = (9*n-1)*a(n-1)/(n+1) - 10*(2*n-3)*a(n-2)/(n+1), n >= 2, a(0)=1, a(1)=2.

Mathematica

CoefficientList[Series[(1-4*x)*(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) - 3*Catalan(n-1)/(2*n-3): n in [1..40]]; // G. C. Greubel, Jul 28 2024
(SageMath)
@CachedFunction
def A046814(n): return 1 if n==0 else 5*A046814(n-1) - 3*catalan_number(n)/(2*n-1)
[A046814(n) for n in range(41)] # G. C. Greubel, Jul 28 2024

Crossrefs

Cf. A000108, A046527, A046714, A046748.

Sequence in context: A183908 A280119 A224032 * A361698 A305547 A007857

Adjacent sequences: A046811 A046812 A046813 * A046815 A046816 A046817

Keyword

easy,nonn

Author

Wolfdieter Lang

Extensions

Offset corrected by Sean A. Irvine, Apr 25 2021

Status

approved