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Fourth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
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%I #28 Oct 05 2022 03:24:12

%S 1,10,70,400,2020,9352,40600,167680,665440,2555840,9551936,34880000,

%T 124853120,439228160,1521839360,5202292736,17571249920,58712184320,

%U 194280061440,637228462080,2073332481024,6696470231040

%N Fourth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

%H Muniru A Asiru, <a href="/A073391/b073391.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-30,0,120,-48,-240,0,240,160,32).

%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073390(k).

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

%F a(n) = (2*(419 + 326*n + 79*n^2 + 6*n^3)*(n+1)*U(n+1) + (458 + 421*n + 112*n^2 + 9*n^3)*(n+2)*U(n))/(2^5*3^4), with U(n) = A002605(n), n >= 0.

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

%t CoefficientList[Series[1/(1-2*x-2*x^2)^5, {x,0,40}], x] (* _G. C. Greubel_, Oct 04 2022 *)

%o (GAP) List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+4,4)*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)^5 )); // _G. C. Greubel_, Oct 04 2022

%o (SageMath)

%o def A073391_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( 1/(1-2*x-2*x^2)^5 ).list()

%o A073391_list(40) # _G. C. Greubel_, Oct 04 2022

%Y Fifth (m=4) column of triangle A073387.

%Y Cf. A002605, A073390.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 02 2002