cp's OEIS Frontend

A073391 Fourth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

%I A073391 #28 Oct 05 2022 03:24:12
%S A073391 1,10,70,400,2020,9352,40600,167680,665440,2555840,9551936,34880000,
%T A073391 124853120,439228160,1521839360,5202292736,17571249920,58712184320,
%U A073391 194280061440,637228462080,2073332481024,6696470231040
%N A073391 Fourth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
%H A073391 Muniru A Asiru, <a href="/A073391/b073391.txt">Table of n, a(n) for n = 0..500</a>
%H A073391 <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 A073391 a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073390(k).
%F A073391 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+4, 4)*binomial(n-k, k)*2^(n-k).
%F A073391 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 A073391 G.f.: 1/(1-2*x*(1+x))^5.
%t A073391 CoefficientList[Series[1/(1-2*x-2*x^2)^5, {x,0,40}], x] (* _G. C. Greubel_, Oct 04 2022 *)
%o A073391 (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 A073391 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^5 )); // _G. C. Greubel_, Oct 04 2022
%o A073391 (SageMath)
%o A073391 def A073391_list(prec):
%o A073391     P.<x> = PowerSeriesRing(ZZ, prec)
%o A073391     return P( 1/(1-2*x-2*x^2)^5 ).list()
%o A073391 A073391_list(40) # _G. C. Greubel_, Oct 04 2022
%Y A073391 Fifth (m=4) column of triangle A073387.
%Y A073391 Cf. A002605, A073390.
%K A073391 nonn,easy
%O A073391 0,2
%A A073391 _Wolfdieter Lang_, Aug 02 2002