login
Fibonomial coefficients.
2

%I #24 May 08 2016 11:16:17

%S 1,55,4895,352440,27372840,2063912136,157373300370,11948265189630,

%T 908637119420910,69056421075989160,5249543573067466872,

%U 399024295188779925720,30331388438447118520355

%N Fibonomial coefficients.

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

%F 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).

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

%p with(combinat): a:=n->1/2227680*fibonacci(n)*fibonacci(n+1) *fibonacci(n+2) *fibonacci(n+3) *fibonacci(n+4) *fibonacci(n+5) *fibonacci(n+6) *fibonacci(n+7) *fibonacci(n+8): seq(a(n), n=1..13); # _Zerinvary Lajos_, Oct 07 2007

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

%o (PARI) b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));

%o vector(20, n, b(n-1, 9)) \\ _Joerg Arndt_, May 08 2016

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

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 10 2000