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Rectangular array of numbers Fibonacci(m(n+1))/Fibonacci(m), m >= 1, n >= 0, read by downward antidiagonals.
20

%I #41 Oct 31 2019 01:44:16

%S 1,1,1,1,3,2,1,4,8,3,1,7,17,21,5,1,11,48,72,55,8,1,18,122,329,305,144,

%T 13,1,29,323,1353,2255,1292,377,21,1,47,842,5796,15005,15456,5473,987,

%U 34,1,76,2208,24447,104005,166408,105937,23184,2584,55,1,123,5777

%N Rectangular array of numbers Fibonacci(m(n+1))/Fibonacci(m), m >= 1, n >= 0, read by downward antidiagonals.

%C Every integer-valued quotient of two Fibonacci numbers is in this array. - _Clark Kimberling_, Aug 28 2008

%C Not only does 5 divide row 5, but 50 divides (-5 + row 5), as in A214984. - _Clark Kimberling_, Nov 02 2012

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 142.

%H Clark Kimberling, <a href="/A028412/b028412.txt">Table of n, a(n) for n = 0..1829</a>

%H I. Strazdins, <a href="http://dx.doi.org/10.1007/978-94-011-5020-0_44">Lucas factors and a Fibonomial generating function</a>, in Applications of Fibonacci numbers, Vol. 7 (Graz, 1996), 401-404, Kluwer Acad. Publ., Dordrecht, 1998.

%F T(n, m) = Sum_{i_1>=0} Sum_{i_2>=0} ... Sum_{i_m>=0} C(n-i_m, i_1)*C(n-i_1, i_2)*C(n-i_2, i_3)*...*C(n-i_{m-1}, i_m).

%F G.f. for column m >= 1: 1/(1 - Lucas(m)*x + (-1)^m*x^2), where Lucas(m) = A000204(m). - _Paul D. Hanna_, Jan 28 2012

%e 1 1 1 1 1 1

%e 1 3 4 7 11 18

%e 2 8 17 48 122 323

%e 3 21 72 329 1353 5796

%e 5 55 305 2255 15005 104005

%e 8 144 1292 15456 166408 1866294

%e 13 377 5473 105937 1845493 33489287

%e ...

%t max = 11; col[m_] := CoefficientList[ Series[ 1/(1 - LucasL[m]*x + (-1)^m*x^2), {x, 0, max}], x]; t = Transpose[ Table[ col[m], {m, 1, max}]] ; Flatten[ Table[ t[[n - m + 1, m]], {n, 1, max }, {m, n, 1, -1}]] (* _Jean-François Alcover_, Feb 21 2012, after _Paul D. Hanna_ *)

%t f[n_] := Fibonacci[n]; t[m_, n_] := f[m*n]/f[n]

%t TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]] (* array *)

%t t = Flatten[Table[t[k, n + 1 - k], {n, 1, 120}, {k, 1, n}]] (* sequence *) (* _Clark Kimberling_, Nov 02 2012 *)

%o (PARI) {T(n,m)=polcoeff(1/(1 - Lucas(m)*x + (-1)^m*x^2 +x*O(x^n)),n)}

%Y Columns include A000045, A001906, A001076, A004187, A049666, A049660, A049667, A049668, A049669, A049670.

%Y Rows include (essentially) A000032, A047946, A083564, A103226.

%Y Main diagonal is A051294.

%Y Transpose is A214978.

%K nonn,tabl,easy,nice

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _Erich Friedman_, Jun 03 2001

%E Edited by _Ralf Stephan_, Feb 03 2005

%E Better description from _Clark Kimberling_, Aug 28 2008