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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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15
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Every integer-valued quotient of two Fibonacci numbers is in this array. [From Clark Kimberling (ck6(AT)evansville.edu), Aug 28 2008]
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REFERENCES
| 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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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). [From Paul D. Hanna, Jan 28 2012]
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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,...
...
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PROG
| (PARI) {T(n, m)=polcoeff(1/(1 - Lucas(m)*x + (-1)^m*x^2 +x*O(x^n)), n)}
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CROSSREFS
| Columns include A000045, A001906, A001076, A004187, A049666, A049660, A049667, A049668, A049669, A049670. Rows include (essentially) A000032, A047946, A083564, A103226. Main diagonal is A051294.
Sequence in context: A092486 A159966 A119263 * A156699 A077819 A030313
Adjacent sequences: A028409 A028410 A028411 * A028413 A028414 A028415
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KEYWORD
| tabl,nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Erich Friedman (efriedma(AT)stetson.edu), Jun 03 2001
Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Feb 03 2005
Better description from Clark Kimberling, Aug 28 2008
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