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A208588 Row square-sums of triangle A185384.

%I A208588 #15 Dec 07 2021 03:35:35
%S A208588 1,5,65,979,15345,247535,4069155,67773805,1139789185,19311870095,
%T A208588 329149434263,5637030686105,96925730626035,1672193347218577,
%U A208588 28932082285914005,501821453320612915,8722842168045249345,151912536408383664095,2650102280875677625415
%N A208588 Row square-sums of triangle A185384.
%H A208588 Vincenzo Librandi, <a href="/A208588/b208588.txt">Table of n, a(n) for n = 0..200</a>
%F A208588 a(n) = sum(binomial(n,k)^2*F(2n-k+1)^2, k=0..n), where F(n) are the Fibonacci numbers (A000045).
%F A208588 Recurrence: (n-2)*(n-1)*n*(1305*n^4 - 14094*n^3 + 57321*n^2 - 104304*n + 71944)*a(n) = 4*(n-2)*(n-1)*(6525*n^5 - 74385*n^4 + 327843*n^3 - 685683*n^2 + 653864*n - 201144)*a(n-1) - 6*(n-2)*(9135*n^6 - 125193*n^5 + 695046*n^4 - 1997365*n^3 + 3132821*n^2 - 2544304*n + 838848)*a(n-2) + 4*(32625*n^7 - 514170*n^6 + 3412449*n^5 - 12326760*n^4 + 26068504*n^3 - 32079108*n^2 + 21061664*n - 5595312)*a(n-3) - 3*(n-3)*(66555*n^6 - 918894*n^5 + 5135607*n^4 - 14853260*n^3 + 23457868*n^2 - 19205728*n + 6392000)*a(n-4) + 16*(n-4)*(n-3)*(6525*n^5 - 73080*n^4 + 309312*n^3 - 634788*n^2 + 644096*n - 264112)*a(n-5) - 4*(n-5)*(n-4)*(n-3)*(1305*n^4 - 8874*n^3 + 22869*n^2 - 26724*n + 12172)*a(n-6). - _Vaclav Kotesovec_, Aug 13 2013
%F A208588 a(n) ~ sqrt(58+26*sqrt(5)) * (9+4*sqrt(5))^n/(20*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 13 2013
%F A208588 Equivalently, a(n) ~ phi^(6*n + 7/2) / (10*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 07 2021
%t A208588 Table[Sum[Binomial[n,k]^2Fibonacci[2n-k+1]^2,{k,0,n}],{n,0,20}]
%o A208588 (Maxima) makelist(sum(binomial(n,k)^2*fib(2*n-k+1)^2,k,0,n),n,0,20);
%Y A208588 Cf. A185384.
%K A208588 nonn,easy
%O A208588 0,2
%A A208588 _Emanuele Munarini_, Feb 29 2012