# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a001657 Showing 1-1 of 1 %I A001657 M4568 N1945 #67 Apr 13 2022 13:25:16 %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, Table of n, a(n) for n = 0..200 %H A001657 A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83. %H A001657 Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 17. %H A001657 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. %H A001657 Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 %H A001657 Index entries for linear recurrences with constant coefficients, 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 %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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE