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Row sums of Fibonomial triangle A010048.
7

%I #34 Jun 03 2024 12:26:37

%S 1,2,3,6,14,42,158,756,4594,35532,349428,4370436,69532964,1407280392,

%T 36228710348,1186337370456,49415178236344,2618246576596392,

%U 176462813970065208,15128228719573952976,1649746715671916095304

%N Row sums of Fibonomial triangle A010048.

%H 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 a(n) = Sum_{m=0..n} A010048(n, m), where A010048(n, m) = fibonomial(n, m).

%F From _Vaclav Kotesovec_, Apr 30 2015: (Start)

%F a(n) ~ c * ((1+sqrt(5))/2)^(n^2/4), where

%F c = EllipticTheta[3,0,1/GoldenRatio] / QPochhammer[-1/GoldenRatio^2] = 2.082828701647012450835512317685120373906427048806222527375... if n is even,

%F c = EllipticTheta[2,0,1/GoldenRatio] / QPochhammer[-1/GoldenRatio^2] = 2.082828691334156222136965926255238646603356514964103252122... if n is odd.

%F 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 (End)

%t 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 (* 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 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 (Maxima) ffib(n):=prod(fib(k),k,1,n);

%o fibonomial(n,k):=ffib(n)/(ffib(k)*ffib(n-k));

%o makelist(sum(fibonomial(n,k),k,0,n),n,0,30); /* _Emanuele Munarini_, Apr 02 2012 */

%Y Cf. A010048, A062073, A181926.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 10 2000