A056565 - OEIS

%I #38 Sep 08 2022 08:45:01

%S 1,21,714,19635,582505,16776144,488605194,14169550626,411591708660,

%T 11948265189630,346934172869802,10072785423545712,292460526776698763,

%U 8491396839675395415,246543315138161480670,7158243695757340957617

%N Fibonomial coefficients.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (21,273,-1092,-1820,1092,273,-21,-1).

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

%F 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).

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

%p with(combinat):

%p a:= n-> 1/3120 *fibonacci(n) *fibonacci(n+1) *fibonacci(n+2) *fibonacci(n+3) *fibonacci(n+4) *fibonacci(n+5) *fibonacci(n+6):

%p seq(a(n), n=1..17); # _Zerinvary Lajos_, Oct 07 2007

%t (Times@@@Partition[Fibonacci[Range[30]],7,1])/3120  (* _Harvey P. Dale_, Apr 10 2011 *)

%o (Magma) [ &*[Fibonacci(n+k): k in [0..6]]/3120: n in [1..16] ]; // _Bruno Berselli_, Apr 11 2011

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

%o vector(20, n, b(n-1, 7))  \\ _Joerg Arndt_, May 08 2016

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

%K nonn,easy

%O 1,2

%A _Wolfdieter Lang_, Jul 10 2000