A073397 - OEIS

%I #17 Aug 20 2024 23:16:30

%S 1,18,198,1680,12060,76824,446952,2420352,12363120,60151520,280833696,

%T 1265442048,5528697408,23507763840,97575960960,396398370816,

%U 1579498956288,6184543546368,23833455191040,90522348871680,339263015528448,1255995653197824,4597442198728704

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

%C For a(n) in terms of U(n+1) and U(n), with U(n) = A002605(n), see A073387 and the row polynomials of triangles A073405 and A073406.

%H G. C. Greubel, <a href="/A073397/b073397.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (18,-126,384,-144,-2016,3360,4608,-12384,-8512, 24768,18432,-26880,-32256,4608,24576,16128,4608,512).

%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073394(k).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+8, 8)*binomial(n-k, k)*2^(n-k).

%F G.f.: 1/(1-2*x*(1+x))^9.

%t CoefficientList[Series[1/(1-2*x-2*x^2)^9, {x,0,30}], x] (* _G. C. Greubel_, Oct 06 2022 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^9 )); // _G. C. Greubel_, Oct 06 2022

%o (SageMath)

%o def A073397_list(prec):

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

%o     return P( 1/(1-2*x-2*x^2)^9 ).list()

%o A073397_list(30) # _G. C. Greubel_, Oct 06 2022

%Y Ninth (m=8) column of triangle A073387.

%Y Cf. A002605, A073387, A073394, A073405, A073406.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 02 2002