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 A028412 Rectangular array of numbers Fibonacci(m(n+1))/Fibonacci(m), m >= 1, n >= 0, read by antidiagonals. 20
 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, 13, 1, 29, 323, 1353, 2255, 1292, 377, 21, 1, 47, 842, 5796, 15005, 15456, 5473, 987, 34, 1, 76, 2208, 24447, 104005, 166408, 105937, 23184, 2584, 55, 1, 123, 5777 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Every integer-valued quotient of two Fibonacci numbers is in this array. - Clark Kimberling, Aug 28 2008 Not only does 5 divide row 5, but 50 divides (-5 + row 5), as in A214984. - Clark Kimberling, Nov 02 2012 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 142. LINKS Clark Kimberling, Table of n, a(n) for n = 0..1829 I. Strazdins, Lucas factors and a Fibonomial generating function, in Applications of Fibonacci numbers, Vol. 7 (Graz, 1996), 401-404, Kluwer Acad. Publ., Dordrecht, 1998. FORMULA 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). 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 EXAMPLE 1   1    1      1       1        1    1   3    4      7      11       18    2   8   17     48     122      323    3  21   72    329    1353     5796    5  55  305   2255   15005   104005    8 144 1292  15456  166408  1866294   13 377 5473 105937 1845493 33489287   ... MATHEMATICA 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 *) f[n_] := Fibonacci[n]; t[m_, n_] := f[m*n]/f[n] TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]] (* array *) t = Flatten[Table[t[k, n + 1 - k], {n, 1, 120}, {k, 1, n}]] (* sequence *) (* Clark Kimberling, Nov 02 2012 *) PROG (PARI) {T(n, m)=polcoeff(1/(1 - Lucas(m)*x + (-1)^m*x^2 +x*O(x^n)), n)} CROSSREFS Columns include A000045, A001906, A001076, A004187, A049666, A049660, A049667, A049668, A049669, A049670. Rows include (essentially) A000032, A047946, A083564, A103226. Main diagonal is A051294. Transpose is A214978. Sequence in context: A092486 A159966 A119263 * A156699 A245183 A262347 Adjacent sequences:  A028409 A028410 A028411 * A028413 A028414 A028415 KEYWORD nonn,tabl,easy,nice AUTHOR EXTENSIONS More terms from Erich Friedman, Jun 03 2001 Edited by Ralf Stephan, Feb 03 2005 Better description from Clark Kimberling, Aug 28 2008 STATUS approved

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