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%I #28 Oct 05 2022 03:25:11
%S 1,12,96,616,3444,17472,82432,367488,1565280,6421376,25525248,
%T 98773248,373450112,1383674880,5036089344,18041821184,63727070976,
%U 222249968640,766234140672,2614196680704,8834194123776
%N Fifth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
%H Muniru A Asiru, <a href="/A073392/b073392.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-48,40,180,-288,-384,576,720,-320,-768,-384,-64).
%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073391(k).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+5, 5)*binomial(n-k, k)*2^(n-k).
%F a(n) = (n+4)*(n+8)*((19*n^2 + 158*n + 275)*(n+1)*U(n+1) + 2*(7*n^2 + 52*n + 65)*(n+2)*U(n))/(2^6*3^4*5), with U(n) = A002605(n), n >= 0.
%F G.f.: 1/(1-2*x*(1+x))^6.
%e x^6 + 12*x^7 + 96*x^8 + 616*x^9 + 3444*x^10 + ... + 222249968640*x^23 + 766234140672*x^24 + 2614196680704*x^25 + 8834194123776*x^26 + ... - _Zerinvary Lajos_, Jun 03 2009
%t CoefficientList[Series[1/(1-2*x*(1+x))^6, {x,0,30}],x] (* _Harvey P. Dale_, May 12 2018 *)
%o (Sage) taylor( 1/(1-2*x-2*x^2)^6, x, 0, 30).list() # _Zerinvary Lajos_, Jun 03 2009; modified by _G. C. Greubel_, Oct 04 2022
%o (GAP) List([0..30], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+5,5)*Binomial(n-k,k)*(1/2)^k)); # _Muniru A Asiru_, Jun 12 2018
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^6 )); // _G. C. Greubel_, Oct 04 2022
%Y Sixth (m=5) column of triangle A073387.
%Y Cf. A002605, A073391.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 02 2002