A073383 Sixth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
1, 14, 119, 784, 4396, 22008, 101220, 435696, 1777986, 6943244, 26129950, 95282992, 338108876, 1171554776, 3975215844, 13239402960, 43364985867, 139925413866, 445409413421, 1400429394784, 4353771487912
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (14,-77,196,-161,-238,427,184,-427,-238,161,196,77, 14,1).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^7 )); // G. C. Greubel, Oct 02 2022 -
Mathematica
CoefficientList[Series[1/(1-2*x-x^2)^7, {x,0,70}], x] (* G. C. Greubel, Oct 02 2022 *) LinearRecurrence[{14,-77,196,-161,-238,427,184,-427,-238,161,196,77,14,1},{1,14,119,784,4396,22008,101220,435696,1777986,6943244,26129950,95282992,338108876,1171554776},30] (* Harvey P. Dale, Apr 26 2025 *) -
SageMath
def A073383_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P( 1/(1-2*x-x^2)^7 ).list() A073383_list(40) # G. C. Greubel, Oct 02 2022
Formula
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+6, 6) * binomial(n-k, k).
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.
G.f.: 1/(1-(2+x)*x)^7.
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