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

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A000108, A046527, A046714, A046748.

Programs

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

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 *)

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

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.

Extensions

Offset corrected by Sean A. Irvine, Apr 25 2021