A073373 Third convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

1, 4, 18, 60, 195, 576, 1644, 4488, 11925, 30860, 78278, 195012, 478599, 1159080, 2774880, 6575280, 15439065, 35955540, 83118970, 190862860, 435601611, 988620624, 2232236628, 5016441240, 11224087965

Offset: 0

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,2,-20,-1,40,8,-32,-16).

Crossrefs

Fourth (m=3) column of triangle A073370.

Cf. A001045, A073372.

Programs

Magma

[(1/4374)*(2^(n+4)*(226 +267*n +90*n^2 +9*n^3) +(-1)^n*(758 +555*n +126*n^2 +9*n^3)): n in [0..40]]; // G. C. Greubel, Sep 29 2022

Mathematica

Table[(1/4374)*(2^(n+4)*(226 +267*n +90*n^2 +9*n^3) +(-1)^n*(758 +555*n +126*n^2 +9*n^3)), {n,0,40}] (* G. C. Greubel, Sep 29 2022 *)

SageMath

def A073373(n): return (1/4374)*(2^(n+4)*(226+267*n+90*n^2+9*n^3) +(-1)^n*(758 +555*n+126*n^2+9*n^3))
[A073373(n) for n in range(40)] # G. C. Greubel, Sep 29 2022

Formula

a(n) = Sum_{k=0..n} b(k) * c(n-k), with b(k) = A001045(k+1) and c(k) = A073372(k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, 3) * binomial(n-k, k) * 2^k.

a(n) = ((350+177*n+21*n^2)*(n+1)*U(n+1) + 2*(277+132*n+15*n^2)*(n+2)*U(n))/ (2*9^3) with U(n) = A001045(n+1), n>=0.

G.f.: 1/(1-(1+2*x)*x)^4 = 1/ ( (1+x)^4*(1-2*x)^4 ).

E.g.f.: (1/4374)*(32*(113 + 366*x + 234*x^2 + 36*x^3)*exp(2*x) - (-758 + 690*x - 153*x^2 + 9*x^3)*exp(-x)). - G. C. Greubel, Sep 29 2022