A073383 - OEIS

%I #18 Oct 03 2022 04:45:03

%S 1,14,119,784,4396,22008,101220,435696,1777986,6943244,26129950,

%T 95282992,338108876,1171554776,3975215844,13239402960,43364985867,

%U 139925413866,445409413421,1400429394784,4353771487912

%N Sixth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

%H G. C. Greubel, <a href="/A073383/b073383.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (14,-77,196,-161,-238,427,184,-427,-238,161,196,77, 14,1).

%F a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073382(k).

%F a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+6, 6) * binomial(n-k, k).

%F a(n) = (7*(173205 +212028*n +96812*n^2 +20728*n^3 +2092*n^4 +80*n^5)*(n+1)* U(n+1) + (262125 +435150*n +232364*n^2 +54548*n^3 +5836*n^4 +232*n^5)*(n+2)* U(n) )/(6!*8^4), with U(n) = A000129(n+1), n >= 0.

%F G.f.: 1/(1-(2+x)*x)^7.

%F a(n) = F''''''(n+7, 2)/6!, that is, 1/6! times the 6th derivative of the (n+7)-th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006

%t CoefficientList[Series[1/(1-2*x-x^2)^7, {x,0,70}], x] (* _G. C. Greubel_, Oct 02 2022 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^7 )); // _G. C. Greubel_, Oct 02 2022

%o (SageMath)

%o def A073383_list(prec):

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

%o     return P( 1/(1-2*x-x^2)^7 ).list()

%o A073383_list(40) # _G. C. Greubel_, Oct 02 2022

%Y Seventh (m=6) column of triangle A054456, A073382.

%Y Cf. A000129.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 02 2002