A073374 Fourth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

1, 5, 25, 95, 340, 1106, 3430, 10130, 28915, 80035, 216143, 571225, 1482110, 3783640, 9522740, 23665300, 58149845, 141435985, 340854645, 814589475, 1931900376, 4549699950, 10645737330, 24761578470

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 (5,0,-30,15,81,-30,-120,0,80,32).

Crossrefs

Fifth (m=4) column of triangle A073370.

Cf. A001045, A073373.

Programs

Magma

[(2^(n+5)*(4208+5790*n+2565*n^2+450*n^3+27*n^4) + (-1)^n*(22808+18510*n+ 5265*n^2+630*n^3+27*n^4))/157464: n in [0..40]]; // G. C. Greubel, Sep 29 2022

Mathematica

Table[(2^(n+5)*(4208+5790*n+2565*n^2+450*n^3+27*n^4) + (-1)^n*(22808+18510*n+ 5265*n^2+630*n^3+27*n^4))/157464, {n,0,40}] (* G. C. Greubel, Sep 29 2022 *)

SageMath

def A073374(n): return (2^(n+5)*(4208+5790*n+2565*n^2+450*n^3+27*n^4) + (-1)^n*(22808+18510*n+ 5265*n^2+630*n^3+27*n^4))/157464
[A073374(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) = A073373(k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+4, 4) * binomial(n-k, k) * 2^k.

a(n) = (5*(2968 +1974*n +411*n^2 +27*n^3)*(n+1)*U(n+1) + 2*(9412 +6099*n +1248*n^2 +81*n^3)*(n+2)*U(n))/(4!*3^7) with U(n) = A001045(n+1), n>=0.

G.f.: 1/(1-(1+2*x)*x)^5 = 1/((1+x)*(1-2*x))^5.

E.g.f.: (1/157464)*(512*(263 + 1104*x + 1026*x^2 + 306*x^3 + 27*x^4)*exp(2*x) + (22808 - 24432*x + 7344*x^2 - 792*x^3 + 27*x^4)*exp(-x)). - G. C. Greubel, Sep 29 2022