A056569 Row sums of Fibonomial triangle A010048.

1, 2, 3, 6, 14, 42, 158, 756, 4594, 35532, 349428, 4370436, 69532964, 1407280392, 36228710348, 1186337370456, 49415178236344, 2618246576596392, 176462813970065208, 15128228719573952976, 1649746715671916095304

Offset: 0

Author

Wolfdieter Lang, Jul 10 2000

Keywords

• nonn
• easy

Links

• S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.

Crossrefs

Cf. A010048, A062073, A181926.

Programs

Mathematica

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 *)
(* Or, since version 10 *) Table[Sum[Fibonorial[n]/Fibonorial[k]/Fibonorial[n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Apr 30 2015 *)
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 *)

Maxima

ffib(n):=prod(fib(k),k,1,n);
fibonomial(n,k):=ffib(n)/(ffib(k)*ffib(n-k));
makelist(sum(fibonomial(n,k),k,0,n),n,0,30); /* Emanuele Munarini, Apr 02 2012 */

Formula

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

From Vaclav Kotesovec, Apr 30 2015: (Start)

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

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

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

Or c = Sum_{j} ((1+sqrt(5))/2)^(-(j+(1-(-1)^n)/4)^2) / A062073, where A062073 = 1.2267420107203532444176302... is the Fibonacci factorial constant.

(End)