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A028412
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Rectangular array of numbers Fibonacci(m(n+1))/Fibonacci(m), m>=1, n>=0, read by antidiagonals.
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20
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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
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graph;
refs;
listen;
history;
text;
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OFFSET
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0,5
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COMMENTS
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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 (-1 + row 5), as in A214984. [Clark Kimberling_, Nov 02 2012]
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 142.
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.
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LINKS
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Clark Kimberling, Table of n, a(n) for n = 0..1829
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FORMULA
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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]
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EXAMPLE
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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,...
...
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MATHEMATICA
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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}]] (* From 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 *)
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PROG
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(PARI) {T(n, m)=polcoeff(1/(1 - Lucas(m)*x + (-1)^m*x^2 +x*O(x^n)), n)}
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CROSSREFS
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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 A182236 A077819
Adjacent sequences: A028409 A028410 A028411 * A028413 A028414 A028415
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KEYWORD
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tabl,nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Erich Friedman, Jun 03 2001
Edited by Ralf Stephan, Feb 03 2005
Better description from Clark Kimberling, Aug 28 2008
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STATUS
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approved
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