A073393 Sixth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.
1, 14, 126, 896, 5488, 30240, 153888, 735744, 3344544, 14581952, 61378240, 250693632, 997593856, 3880249856, 14791776768, 55385874432, 204082373376, 741186464256, 2656771815936, 9410113241088
Offset: 0
Examples
x^7 + 14*x^8 + 126*x^9 + 896*x^10 + 5488*x^11 + ... + 204082373376*x^23 + 741186464256*x^24 + 2656771815936*x^25 + 9410113241088*x^26 + ... - _Zerinvary Lajos_, Jun 03 2009
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (14,-70,112,196,-728,-168,1920,336,-2912,-1568, 1792,2240,896,128).
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^7 )); // G. C. Greubel, Oct 05 2022 -
Mathematica
CoefficientList[Series[1/(1-2x(1+x))^7,{x,0,30}],x] (* or *) LinearRecurrence[{14,-70,112,196,-728,-168,1920,336,-2912,-1568,1792,2240,896,128},{1,14,126,896,5488,30240,153888,735744,3344544,14581952,61378240,250693632, 997593856,3880249856},30](* Harvey P. Dale, Jan 24 2013 *) -
SageMath
taylor( 1/(1-2*x-2*x^2)^7, x, 0, 26).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 05 2022
Formula
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+6, 6)*binomial(n-k, k)*2^(n-k).
a(n) = ((54340 + 59802*n + 24583*n^2 + 4747*n^3 + 433*n^4 + 15*n^5)*(n+1)*U(n+1) + (23420 + 32768*n + 15333*n^2 + 3201*n^3 + 307*n^4 + 11*n^5)*(n+2)*U(n))/(2^7*3^5*5), with U(n) := A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^7.
a(0)=1, a(1)=14, a(2)=126, a(3)=896, a(4)=5488, a(5)=30240, a(6)=153888, a(7)=735744, a(8)=3344544, a(9)=14581952, a(10)=61378240, a(11)=250693632, a(12)=997593856, a(13)=3880249856, a(n) = 14*a(n-1) - 70*a(n-2) + 112*a(n-3) + 196*a(n-4) - 728*a(n-5) - 168*a(n-6) + 1920*a(n-7) + 336*a(n-8) - 2912*a(n-9) - 1568*a(n-10) + 1792*a(n-11) + 2240*a(n-12) + 896*a(n-13) + 128*a(n-14). - Harvey P. Dale, Jan 24 2013