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A056569 Row sums of Fibonomial triangle A010048. 7
1, 2, 3, 6, 14, 42, 158, 756, 4594, 35532, 349428, 4370436, 69532964, 1407280392, 36228710348, 1186337370456, 49415178236344, 2618246576596392, 176462813970065208, 15128228719573952976, 1649746715671916095304 (list; graph; refs; listen; history; text; internal format)
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(A010048(n, m), m=0..n); 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: A007611 A098641 A188775 * A094468 A091285 A109459

Adjacent sequences:  A056566 A056567 A056568 * A056570 A056571 A056572

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jul 10 2000

STATUS

approved

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Last modified November 23 13:09 EST 2017. Contains 295127 sequences.