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A001657 Fibonomial coefficients: column 5 of A010048.

%I A001657 M4568 N1945 #77 May 07 2025 11:14:31
%S A001657 1,8,104,1092,12376,136136,1514513,16776144,186135312,2063912136,
%T A001657 22890661872,253854868176,2815321003313,31222272414424,
%U A001657 346260798314872,3840089017377228,42587248616222024,472299787252290712,5237885063192296801,58089034826620525728
%N A001657 Fibonomial coefficients: column 5 of A010048.
%D A001657 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001657 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001657 T. D. Noe, <a href="/A001657/b001657.txt">Table of n, a(n) for n = 0..200</a>
%H A001657 A. Brousseau, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/6-1/brousseau3.pdf">A sequence of power formulas</a>, Fib. Quart., 6 (1968), 81-83.
%H A001657 Alfred Brousseau, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972. See p. 17.
%H A001657 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001657 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001657 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,40,-60,-40,8,1).
%F A001657 a(n) = A010048(5+n, 5) (or fibonomial(5+n, 5)).
%F A001657 G.f.: 1/(1-8*x-40*x^2+60*x^3+40*x^4-8*x^5-x^6) = 1/((1-x-x^2)*(1+4*x-x^2)*(1-11*x-x^2)) (see Comments to A055870).
%F A001657 a(n) = 11*a(n-1) + a(n-2) + ((-1)^n)*fibonomial(n+3, 3), n >= 2; a(0)=1, a(1)=8; fibonomial(n+3, 3)= A001655(n).
%F A001657 a(n) = Fibonacci(n+3)*(Fibonacci(n+3)^4-1)/30. - _Gary Detlefs_, Apr 24 2012
%F A001657 a(n) = (A049666(n+3) + 2*(-1)^n*A001076(n+3) - 3*A000045(n+3))/150, n >= 0, with A049666(n) = F(5*n)/5, A001076(n) = F(3*n)/2 and A000045(n) = F(n). From the partial fraction decomposition of the o.g.f. and recurrences. - _Wolfdieter Lang_, Aug 23 2012
%F A001657 a(n) = a(-6-n) * (-1)^n for all n in Z. - _Michael Somos_, Sep 19 2014
%F A001657 0 = a(n)*(-a(n+1) - 3*a(n+2)) + a(n+1)*(-8*a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Sep 19 2014
%F A001657 G.f.: exp( Sum_{k>=1} F(6*k)/F(k) * x^k/k ), where F(n) = A000045(n). - _Seiichi Manyama_, May 07 2025
%e A001657 G.f. = 1 + 8*x + 104*x^2 + 1092*x^3 + 12376*x^4 + 136136*x^5 + 1514513*x^6 + ...
%p A001657 with(combinat) : a:=n-> 1/30*fibonacci(n)*fibonacci(n+1)*fibonacci(n+2)*fibonacci(n+3)*fibonacci(n+4): seq(a(n), n=1..19); # _Zerinvary Lajos_, Oct 07 2007
%p A001657 A001657:=-1/(z**2+11*z-1)/(z**2-4*z-1)/(z**2+z-1); # _Simon Plouffe_ in his 1992 dissertation
%t A001657 f[n_] := Times @@ Fibonacci[Range[n + 1, n + 5]]/30; t = Table[f[n], {n, 0, 20}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2010 *)
%t A001657 LinearRecurrence[{8,40,-60,-40,8,1},{1,8,104,1092,12376,136136},20] (* _Harvey P. Dale_, Nov 30 2019 *)
%o A001657 (PARI) a(n)=(n->(n^5-n)/30)(fibonacci(n+3)) \\ _Charles R Greathouse IV_, Apr 24 2012
%o A001657 (PARI) b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
%o A001657 vector(20, n, b(n-1, 5))  \\ _Joerg Arndt_, May 08 2016
%Y A001657 Cf. A010048, A001654-A001658, A065563.
%K A001657 nonn,easy
%O A001657 0,2
%A A001657 _N. J. A. Sloane_
%E A001657 Corrected and extended by _Wolfdieter Lang_, Jun 27 2000