A056568 Fibonomial coefficients.

1, 89, 12816, 1493064, 187628376, 22890661872, 2824135408458, 346934172869802, 42689423937884208, 5249543573067466872, 645693859487298425256, 79413089729752455762384, 9767258556969762111163771, 1201288963378036364032704659, 147748983166877427393815516256

Offset: 0

Author

Wolfdieter Lang, Jul 10 2000

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A010048, A000045, A001654-8, A056565-7, A001906, A004187 (signed), A049660, A049668 (signed), A049670.

Programs

Magma

[&*[Fibonacci(n+i): i in [0..9]]/122522400: n in [1..15]]; // Vincenzo Librandi, Oct 31 2014

Maple

F:= combinat[fibonacci]: a:= n-> mul(F(n+i), i=0..9)/122522400: seq(a(n), n=1..18); # Zerinvary Lajos, Oct 07 2007
a:= n-> (Matrix(11, (i,j)-> if (i=j-1) then 1 elif j=1 then [1514513, -582505, -83215, 4895, 89, -1][abs(i-11/2)+1/2] else 0 fi)^n)[1, 1]; seq(a(n), n=0..18);  # Alois P. Heinz, Aug 15 2008

Mathematica

Times@@@Partition[Fibonacci[Range[30]],10,1]/122522400 (* Harvey P. Dale, Jul 27 2019 *)

PARI

a(n)=prod(k=0,9,fibonacci(n+k))/122522400; \\ Joerg Arndt, Oct 31 2014

Formula

a(n) = A010048(n+10,10) =: Fibonomial(n+10,10).

G.f.: 1/p(11,n) with p(11,n) = 1-89*x -4895*x^2 +83215*x^3 +582505*x^4 -1514513*x^5 -1514513*x^6 +582505*x^7 +83215*x^8 -4895*x^9 -89*x^10 +x^11 = (1+x) *(1-3*x+x^2) *(1+7*x+x^2) *(1-18*x+x^2) *(1+47*x+x^2) *(1-123*x+x^2) (n=8 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).

Recursion: a(n)=123*a(n-1)-a(n-2)+((-1)^n)*A056566(n), n >= 2, a(0)=1, a(1)=89.

G.f.: exp( Sum_{k>=1} F(11*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025