A056565 Fibonomial coefficients.

1, 21, 714, 19635, 582505, 16776144, 488605194, 14169550626, 411591708660, 11948265189630, 346934172869802, 10072785423545712, 292460526776698763, 8491396839675395415, 246543315138161480670, 7158243695757340957617, 207835653079349665473587

Offset: 0

Author

Wolfdieter Lang, Jul 10 2000

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (21,273,-1092,-1820,1092,273,-21,-1).

Crossrefs

Cf. A010048, A000045, A001654-8, A001076, A049666 (signed), A049667.

Programs

Magma

[ &*[Fibonacci(n+k): k in [0..6]]/3120: n in [1..16] ]; // Bruno Berselli, Apr 11 2011

Mathematica

(Times@@@Partition[Fibonacci[Range[30]],7,1])/3120  (* Harvey P. Dale, Apr 10 2011 *)

PARI

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 7))  \\ Joerg Arndt, May 08 2016

Formula

a(n) = A010048(n+7, 7) =: Fibonomial(n+7, 7).

G.f.: 1/p(8, n) with p(8, n) = 1 - 21*x - 273*x^2 + 1092*x^3 + 1820*x^4 - 1092*x^5 - 273*x^6 + 21*x^7 + x^8 = (1 + x - x^2) * (1 - 4*x - x^2) * (1 + 11*x - x^2) * (1 - 29*x - x^2) (n=8 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).

a(n) = 29*a(n-1) + a(n-2) + ((-1)^n) * A001657(n), n >= 2, a(0)=1, a(1)=21.

G.f.: exp( Sum_{k>=1} F(8*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

Extensions

Offset corrected by Seiichi Manyama, May 07 2025