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A056569 Row sums of Fibonomial triangle A010048.

%I A056569 #34 Jun 03 2024 12:26:37
%S A056569 1,2,3,6,14,42,158,756,4594,35532,349428,4370436,69532964,1407280392,
%T A056569 36228710348,1186337370456,49415178236344,2618246576596392,
%U A056569 176462813970065208,15128228719573952976,1649746715671916095304
%N A056569 Row sums of Fibonomial triangle A010048.
%H A056569 S. Falcon, <a href="https://www.saspublishers.com/article/2554/">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
%F A056569 a(n) = Sum_{m=0..n} A010048(n, m), where A010048(n, m) = fibonomial(n, m).
%F A056569 From _Vaclav Kotesovec_, Apr 30 2015: (Start)
%F A056569 a(n) ~ c * ((1+sqrt(5))/2)^(n^2/4), where
%F A056569 c = EllipticTheta[3,0,1/GoldenRatio] / QPochhammer[-1/GoldenRatio^2] = 2.082828701647012450835512317685120373906427048806222527375... if n is even,
%F A056569 c = EllipticTheta[2,0,1/GoldenRatio] / QPochhammer[-1/GoldenRatio^2] = 2.082828691334156222136965926255238646603356514964103252122... if n is odd.
%F A056569 Or c = Sum_{j} ((1+sqrt(5))/2)^(-(j+(1-(-1)^n)/4)^2) / A062073, where A062073 = 1.2267420107203532444176302... is the Fibonacci factorial constant.
%F A056569 (End)
%t A056569 Table[Sum[Product[Fibonacci[j],{j,1,n}] / Product[Fibonacci[j],{j,1,k}] / Product[Fibonacci[j],{j,1,n-k}],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Apr 30 2015 *)
%t A056569 (* Or, since version 10 *) Table[Sum[Fibonorial[n]/Fibonorial[k]/Fibonorial[n-k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Apr 30 2015 *)
%t A056569 Round@Table[Sum[GoldenRatio^(k(n-k)) QBinomial[n, k, -1/GoldenRatio^2], {k, 0, n}], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - _Vladimir Reshetnikov_, Sep 25 2016 *)
%o A056569 (Maxima) ffib(n):=prod(fib(k),k,1,n);
%o A056569 fibonomial(n,k):=ffib(n)/(ffib(k)*ffib(n-k));
%o A056569 makelist(sum(fibonomial(n,k),k,0,n),n,0,30); /* _Emanuele Munarini_, Apr 02 2012 */
%Y A056569 Cf. A010048, A062073, A181926.
%K A056569 nonn,easy
%O A056569 0,2
%A A056569 _Wolfdieter Lang_, Jul 10 2000