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A001658 Fibonomial coefficients.

%I A001658 M4919 N2112 #79 May 07 2025 11:13:20
%S A001658 1,13,273,4641,85085,1514513,27261234,488605194,8771626578,
%T A001658 157373300370,2824135408458,50675778059634,909348684070099,
%U A001658 16317540120588343,292806787575013635,5254201798026392211,94282845030238533383,1691836875411111866723,30358781826262552258596
%N A001658 Fibonomial coefficients.
%C A001658 It appears that a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 6. - _John W. Layman_, Apr 14 2000
%C A001658 Layman's formula is correct. - _Wolfdieter Lang_, Jul 13 2000
%C A001658 Layman's formula is a consequence of formula 2.8 (p. 116) of Lind (1971). - _Dale Gerdemann_, May 08 2016
%D A001658 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001658 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001658 T. D. Noe, <a href="/A001658/b001658.txt">Table of n, a(n) for n = 0..200</a>
%H A001658 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 A001658 A. 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, p. 74.
%H A001658 D. A. Lind, A Determinant Involving Binomial Coefficients, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/9-2/lind-a.pdf">Part 1</a>, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/9-2/lind-b.pdf">Part 2</a>, Fibonacci Quarterly 9.2, 1971.
%H A001658 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (13, 104, -260, -260, 104, 13, -1).
%F A001658 From _Wolfdieter Lang_, Jul 13 2000: (Start)
%F A001658 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).
%F A001658 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)
%F A001658 From _Gary Detlefs_, Dec 03 2012: (Start)
%F A001658 a(n) = F(n+1)*F(n+2)*F(n+3)*F(n+4)*F(n+5)*F(n+6)/240.
%F A001658 a(n) = (F(n+5)^2 - F(n+4)^2)*(F(n+3)^4 - 1)/240, where F(n) = A000045(n). (End)
%F A001658 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
%F A001658 G.f.: exp( Sum_{k>=1} F(7*k)/F(k) * x^k/k ), where F(n) = A000045(n). - _Seiichi Manyama_, May 07 2025
%t A001658 f[n_] := Times @@ Fibonacci[Range[n+1, n+6]]/240; Table[f[n], {n, 0, 20}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2010 *)
%t A001658 LinearRecurrence[{13,104,-260,-260,104,13,-1},{1,13,273,4641,85085,1514513,27261234},20] (* _Harvey P. Dale_, Aug 24 2014 *)
%o A001658 (PARI) b(n,k)=prod(j=1,k,fibonacci(n+j)/fibonacci(j));
%o A001658 vector(20,n,b(n-1,6))  \\ _Joerg Arndt_, May 08 2016
%Y A001658 Cf. A001654, A001656, A001657.
%K A001658 nonn,easy
%O A001658 0,2
%A A001658 _N. J. A. Sloane_
%E A001658 More terms from _Wolfdieter Lang_, Jul 13 2000