A073384 Seventh convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
1, 16, 152, 1104, 6756, 36624, 181224, 834768, 3628746, 15035504, 59829704, 229977904, 857894388, 3117321456, 11067753144, 38492230704, 131417200419, 441252045408, 1459330704656, 4760342849504
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (16,-104,336,-476,-112,1064,-432,-1222,432,1064, 112,-476,-336,-104,-16,-1).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^8 )); // G. C. Greubel, Oct 03 2022 -
Mathematica
CoefficientList[Series[1/(1-2*x-x^2)^8, {x,0,40}], x] (* G. C. Greubel, Oct 03 2022 *) -
SageMath
def A073384_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P( 1/(1-2*x-x^2)^8 ).list() A073384_list(40) # G. C. Greubel, Oct 03 2022
Formula
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+7, 7) * binomial(n-k, k).
a(n) = ((34083315 +46659654*n +24858030*n^2 +6632968*n^3 +939632*n^4 +67304*n^5 + 1912*n^6)*(n+1)*U(n+1) + (7204365 +13225068*n +8230910*n^2 +2411744*n^3 + 362968*n^4 +27088*n^5 +792*n^6)*(n+2)*U(n))/(2^18*3^2*5*7), with U(n) = A000129(n+1), n >= 0.
G.f.: 1/(1-(2+x)*x)^8.
a(n) = F'''''''(n+8, 2)/7!, that is, 1/7! times the 7th derivative of the (n+8)-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006