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

0,2

Links

Table of n, a(n) for n=0..20.

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.

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)

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 *)

Prog

(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

Crossrefs

Cf. A010048, A062073, A181926.

Sequence in context: A098641 A188775 A015892 * A094468 A091285 A109459

Adjacent sequences: A056566 A056567 A056568 * A056570 A056571 A056572

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 10 2000

Status

approved