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A056568 Fibonomial coefficients.

%I A056568 #48 May 07 2025 11:18:14
%S A056568 1,89,12816,1493064,187628376,22890661872,2824135408458,
%T A056568 346934172869802,42689423937884208,5249543573067466872,
%U A056568 645693859487298425256,79413089729752455762384,9767258556969762111163771,1201288963378036364032704659,147748983166877427393815516256
%N A056568 Fibonomial coefficients.
%H A056568 Alois P. Heinz, <a href="/A056568/b056568.txt">Table of n, a(n) for n = 0..150</a>
%F A056568 a(n) = A010048(n+10,10) =: Fibonomial(n+10,10).
%F A056568 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).
%F A056568 Recursion: a(n)=123*a(n-1)-a(n-2)+((-1)^n)*A056566(n), n >= 2, a(0)=1, a(1)=89.
%F A056568 G.f.: exp( Sum_{k>=1} F(11*k)/F(k) * x^k/k ), where F(n) = A000045(n). - _Seiichi Manyama_, May 07 2025
%p A056568 F:= combinat[fibonacci]: a:= n-> mul(F(n+i), i=0..9)/122522400: seq(a(n), n=1..18); # _Zerinvary Lajos_, Oct 07 2007
%p A056568 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
%t A056568 Times@@@Partition[Fibonacci[Range[30]],10,1]/122522400 (* _Harvey P. Dale_, Jul 27 2019 *)
%o A056568 (Magma) [&*[Fibonacci(n+i): i in [0..9]]/122522400: n in [1..15]]; // _Vincenzo Librandi_, Oct 31 2014
%o A056568 (PARI)  a(n)=prod(k=0,9,fibonacci(n+k))/122522400; \\ _Joerg Arndt_, Oct 31 2014
%Y A056568 Cf. A010048, A000045, A001654-8, A056565-7, A001906, A004187 (signed), A049660, A049668 (signed), A049670.
%K A056568 nonn,easy
%O A056568 0,2
%A A056568 _Wolfdieter Lang_, Jul 10 2000