A056567 Fibonomial coefficients.

1, 55, 4895, 352440, 27372840, 2063912136, 157373300370, 11948265189630, 908637119420910, 69056421075989160, 5249543573067466872, 399024295188779925720, 30331388438447118520355, 2305576054220330112077285, 175254358052498673797874685, 13321629629800423781409595728

Offset: 0

Author

Wolfdieter Lang, Jul 10 2000

Keywords

• nonn
• easy

Crossrefs

Cf. A010048, A000045, A001654-8, A056565-6, A001076 (signed), A049666, A049667 (signed), A049669.

Programs

Mathematica

a[n_] := (1/2227680) Times @@ Fibonacci[n + Range[9]]; Array[a, 20, 0] (* Giovanni Resta, May 08 2016 *)

PARI

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 9))  \\ Joerg Arndt, May 08 2016

Formula

a(n) = A010048(n+9, 9) = Fibonomial(n+9, 9).

G.f.: 1/p(10, n) with p(10, n)= 1 - 55*x - 1870*x^2 + 19635*x^3 + 85085*x^4 - 136136*x^5 - 85085*x^6 + 19635*x^7 + 1870*x^8 - 55*x^9 - x^10 = (1 - x - x^2)*(1 + 4*x - x^2)*(1 - 11*x - x^2)*(1 + 29*x - x^2)*(1 - 76*x - x^2) (n=10 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).

Recursion: a(n) = 76*a(n-1) + a(n-2)+((-1)^n)*A056565(n), n >= 2, a(0)=1, a(1)=55.

G.f.: exp( Sum_{k>=1} F(10*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025