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A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

%I A073378 #14 Oct 01 2022 19:18:46
%S A073378 1,9,63,345,1665,7227,29073,109791,394020,1354210,4486482,14397318,
%T A073378 44932446,136817370,407566350,1190446866,3415935699,9645169743,
%U A073378 26836557825,73670997015,199751003991,535449185469
%N A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
%C A073378 For a(n) in terms of U(n+1) and U(n) with U(n) = A001045(n+1) see A073370 and the row polynomials of triangles A073399 and A073400.
%H A073378 G. C. Greubel, <a href="/A073378/b073378.txt">Table of n, a(n) for n = 0..1000</a>
%H A073378 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (9,-18,-60,234,126,-1176,36,3519,-479,-7038,144, 9408,2016,-7488,-3840,2304,2304,512).
%F A073378 a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073377(k).
%F A073378 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+8, 8) * binomial(n-k, k) * 2^k.
%F A073378 G.f.: 1/(1-(1+2*x)*x)^9 = 1/((1+x)*(1-2*x))^9.
%t A073378 CoefficientList[Series[1/((1+x)*(1-2*x))^9, {x,0,40}], x] (* _G. C. Greubel_, Oct 01 2022 *)
%o A073378 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^9 )); // _G. C. Greubel_, Oct 01 2022
%o A073378 (SageMath)
%o A073378 def A073378_list(prec):
%o A073378     P.<x> = PowerSeriesRing(ZZ, prec)
%o A073378     return P( 1/((1+x)*(1-2*x))^9 ).list()
%o A073378 A073378_list(40) # _G. C. Greubel_, Oct 01 2022
%Y A073378 Ninth (m=8) column of triangle A073370.
%Y A073378 Cf. A001045, A073377, A073399, A073400, A073401.
%K A073378 nonn,easy
%O A073378 0,2
%A A073378 _Wolfdieter Lang_, Aug 02 2002