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 A073393 Sixth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself. 1

%I

%S 1,14,126,896,5488,30240,153888,735744,3344544,14581952,61378240,

%T 250693632,997593856,3880249856,14791776768,55385874432,204082373376,

%U 741186464256,2656771815936,9410113241088

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

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (14, -70, 112, 196, -728, -168, 1920, 336, -2912, -1568, 1792, 2240, 896, 128).

%F a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) := A002605(k) and c(k) := A073392(k).

%F a(n) = (2^n)*Sum_{k=0..floor(n/2)} binomial(n-k+6, 6)*binomial(n-k, k)*(1/2)^k.

%F 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.

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

%F 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

%e 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

%t 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 *)

%o (Sage) taylor( mul(x/(1 - 2*x - 2*x^2) for i in range(1,8)),x,0,26) # _Zerinvary Lajos_, Jun 03 2009

%Y Seventh (m=6) column of triangle A073387.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 02 2002

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)