A034009 Convolution of A000295(n+2) (n>=0) with itself.

1, 8, 38, 140, 443, 1268, 3384, 8584, 20965, 49744, 115402, 262996, 590831, 1311900, 2884956, 6293040, 13633305, 29362200, 62916910, 134220380, 285215651, 603983108, 1275072128, 2684358680, 5637149133, 11811165088

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-26,44,-41,20,-4).

Crossrefs

Cf. A000295, A045889.

Programs

Magma

[(16*(n-3)*2^n+(n+7)*(n^2+11*n+42) div 6): n in [0..30]]; // Vincenzo Librandi, Sep 20 2014

Maple

seq(16*(n-3)*2^n+(n+7)*(n^2+11*n+42)/6, n=0..100); # Robert Israel, Sep 19 2014

Mathematica

Table[Sum[ k Binomial[n + 5, k + 4], {k, 0, n+1}], {n, 0, 26}] (* Zerinvary Lajos, Jul 08 2009 *)
Table[(16 (n-3) 2^n + (n + 7) (n^2 + 11 n + 42) / 6), {n, 0, 40}] (* Vincenzo Librandi, Sep 20 2014 *)

Formula

(2^(n+2)-n-3) '*' (2^(n+2)-n-3) where '*' denotes the convolution product.

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

Partial sums of A045889.

a(n) = (n-3)*2^(n+4)+binomial(n+3,3)+4*(binomial(n+1,2)+4*n+12)

= 2^(n+4)*(n-3)+(n+7)*(n*(n+11)+42)/6.

a(n) = binomial(n+3,3)*hypergeom([2,-n],[-n-3],2). - Peter Luschny, Sep 19 2014

a(n) = Sum_{k=0..n+4} Sum_{i=0..n+4} (i-k) * C(n-k+4,i+2). - Wesley Ivan Hurt, Sep 19 2017

Extensions

Edited by Peter Luschny, Sep 20 2014