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 A001658 Fibonomial coefficients. (Formerly M4919 N2112) 6
 1, 13, 273, 4641, 85085, 1514513, 27261234, 488605194, 8771626578, 157373300370, 2824135408458, 50675778059634, 909348684070099, 16317540120588343, 292806787575013635, 5254201798026392211, 94282845030238533383, 1691836875411111866723, 30358781826262552258596 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS It appears a(n) = 13a(n-1) + 104a(n-2) - 260a(n-3) - 260a(n-4) + 104a(n-5) + 13a(n-6) - a(n-7) for n > 6. - John W. Layman, Apr 14 2000 Layman's formula is correct. - Wolfdieter Lang, Jul 13 2000 Layman's formula is a consequence of formula 2.8 (p. 116) of Lind (1971). - Dale Gerdemann, May 08 2016 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..200 A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83. A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 74. D. A. Lind, A Determinant Involving Binomial Coefficients, Part 1, Part 2, Fibonacci Quarterly 9.2, 1971. Index entries for linear recurrences with constant coefficients, signature (13, 104, -260, -260, 104, 13, -1). FORMULA From Wolfdieter Lang, Jul 13 2000: (Start) G.f.: 1/(1-13*x-104*x^2+260*x^3+260*x^4-104*x^5-13*x^6+x^7) = 1/((1+x)*(1-3*x+x^2)*(1+7*x+x^2)*(1-18*x+x^2)) (see Comments to A055870). a(n) = 5*a(n-1)+F(n-5)*Fibonomial(n+5, 5), n >= 1, a(0) = 1; F(n) = A000045(n) (Fibonacci). a(n) = 18*a(n-1)-a(n-2)+((-1)^n)*Fibonomial(n+4, 4), n >= 2; a(0) = 1, a(1) = 13; Fibonomial(n+4, 4) = A001656(n). (End) From Gary Detlefs, Dec 03 2012: (Start) a(n) = F(n+1)*F(n+2)*F(n+3)*F(n+4)*F(n+5)*F(n+6)/240. a(n) = (F(n+5)^2 - F(n+4)^2)*(F(n+3)^4 - 1)/240, where F(n) = A000045(n). (End) Conjecture: a(n) = F(7)^(n-6) + Sum_{i=3..n-5} F(i-2)F(6)^{i-1}F(7)^{n-i-5} + Sum_{j=3..i} F(i-2)F(j-2)F(5)^{j-1}F(6)^{i-j}F(7)^{n-i-5} + Sum_{k=3..j} F(i-2)F(j-2)F(k-2)F(4)^{k-1}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5} + Sum_{l=3..k} F(i-2)F(j-2)F(k-2)F(l-2)F(3)^{l-1}F(4)^{k-l}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5} + Sum_{m=3..l} F(i-2)F(j-2)F(k-2)F(l-2)F(m-2)F(m)F(3)^{l-m}F(4)^{k-l}F(5)^{j-k}F(6)^{i-j}F(7)^{n-i-5}, where F(n)=A000045(n). - Dale Gerdemann, May 08 2016 MAPLE with(combinat):a:=n->1/240*fibonacci(n)*fibonacci(n+1)*fibonacci(n+2)*fibonacci(n+3)*fibonacci(n+4)*fibonacci(n+5): seq(a(n), n=1..17); # Zerinvary Lajos, Oct 07 2007 MATHEMATICA f[n_] := Times @@ Fibonacci[Range[n+1, n+6]]/240; Table[f[n], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *) LinearRecurrence[{13, 104, -260, -260, 104, 13, -1}, {1, 13, 273, 4641, 85085, 1514513, 27261234}, 20] (* Harvey P. Dale, Aug 24 2014 *) PROG (PARI) b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j)); vector(20, n, b(n-1, 6))  \\ Joerg Arndt, May 08 2016 CROSSREFS Cf. A001654, A001656, A001657. Sequence in context: A142262 A163155 A183515 * A034911 A196652 A197455 Adjacent sequences:  A001655 A001656 A001657 * A001659 A001660 A001661 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Wolfdieter Lang, Jul 13 2000 STATUS approved

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