%I #30 Sep 08 2022 08:44:51

%S 1,8,38,140,443,1268,3384,8584,20965,49744,115402,262996,590831,

%T 1311900,2884956,6293040,13633305,29362200,62916910,134220380,

%U 285215651,603983108,1275072128,2684358680,5637149133,11811165088

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

%H Vincenzo Librandi, <a href="/A034009/b034009.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,-26,44,-41,20,-4).

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

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

%F Partial sums of A045889.

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

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

%F a(n) = binomial(n+3,3)*hypergeom([2,-n],[-n-3],2). - _Peter Luschny_, Sep 19 2014

%F 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

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

%t Table[Sum[ k Binomial[n + 5, k + 4], {k, 0, n+1}], {n, 0, 26}] (* _Zerinvary Lajos_, Jul 08 2009 *)

%t Table[(16 (n-3) 2^n + (n + 7) (n^2 + 11 n + 42) / 6), {n, 0, 40}] (* _Vincenzo Librandi_, Sep 20 2014 *)

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

%Y Cf. A000295, A045889.

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_

%E Edited by _Peter Luschny_, Sep 20 2014