A126952 a(0)=1, a(n+1) = 5*a(n)-4*A117641(n) for n>=0.

1, 1, 5, 21, 93, 421, 1937, 9017, 42349, 200277, 952425, 4549953, 21818841, 104966889, 506372277, 2448641061, 11865563853, 57604036309, 280110716777, 1364092539041, 6651682319233, 32474171399649, 158714415664557

Offset

0,3

Comments

Hankel transform is 4^n=A000302(n).

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1001

Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Formula

a(n) = Sum_{k = 0..n} binomial(n,k)*b(k), where b(n) = Sum_{k = 0..n} binomial(n+k,k) * (-2)^(n-k). - Peter Bala, Jun 18 2025

From Vaclav Kotesovec, Jun 22 2025: (Start)

Recurrence: 5*(n-2)*a(n-3) + (9*n-26)*a(n-2) + (12-17*n)*a(n-1) + 3*n*a(n) = 0.

a(n) ~ 5^(n + 1/2) / (4*sqrt(Pi*n)). (End)

Mathematica

Block[{$MaxExtraPrecision = 10^3, s = Rest@ CoefficientList[Series[(1 + 3 x - Sqrt[1 - 6 x + 5 x^2])/(2 x^2 + 6 x), {x, 0, 21}], x]}, Nest[Append[#, 5 #[[-1]] - 4 s[[Length@ # - 1]] ] &, {1, 1}, Length@ s]] (* Michael De Vlieger, Dec 15 2019, after Robert G. Wilson v at A117641 *)

Crossrefs

Cf. A000302, A117641.

Sequence in context: A154964 A007287 A116904 * A273570 A103519 A178876

Adjacent sequences: A126949 A126950 A126951 * A126953 A126954 A126955

Keyword

nonn

Author

Philippe Deléham, Mar 19 2007

Extensions

a(11) and a(22) corrected by Michael De Vlieger, Dec 15 2019

Status

approved