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Search: id:A127836
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| A127836 |
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Triangle read by rows: row n gives coefficients (lowest degree first) of P_n(x), where P_0(x) = P_1(x) = 1; P_n(x) = P_{n-1}(x) + x^(n-1)*P_{n-2}(x). |
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+0
5
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| 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 6, 6, 6, 5, 5
(list; graph; listen)
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OFFSET
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0,17
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COMMENT
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P_n(x) has degree A002620(n).
Row sums are the Fibonacci numbers (A000045). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 12 2007
T(n,k)=number of Fibonacci words of length n-1 in which the sum of the positions of the 0's is equal to k. A Fibonacci binary word is a binary word having no 00 subword. Examples: T(5,4)=2 because we have 1110 and 0101; T(7,6)=3 because we have 111110, 101011 and 011101. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 04 2009]
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REFERENCES
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A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin., 10 (2003), Research Paper 13, 122 pp. See Identity 3-18, pp. 26-27.
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EXAMPLE
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Triangle begins:
1
1
1,1
1,1,1
1,1,1,1,1
1,1,1,1,2,1,1
1,1,1,1,2,2,2,1,1,1
1,1,1,1,2,2,3,2,2,2,2,1,1
1,1,1,1,2,2,3,3,3,3,3,3,3,2,1,1,1
1,1,1,1,2,2,3,3,4,4,4,4,5,4,4,3,3,2,2,1,1
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MAPLE
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P[0]:=1; P[1]:=1; d:=[0, 0]; M:=14; for n from 2 to M do P[n]:=expand(P[n-1]+q^(n-1)*P[n-2]);
lprint(seriestolist(series(P[n], q, M^2))); d:=[op(d), degree(P[n], q)]; od: d;
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CROSSREFS
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Rows converge to A003114 (coefficients in expansion of the first Rogers-Ramanujan identities). Cf. A128915, A119469.
Sequence in context: A002637 A166279 A077478 this_sequence A031262 A047072 A053258
Adjacent sequences: A127833 A127834 A127835 this_sequence A127837 A127838 A127839
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KEYWORD
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nonn,tabf
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Apr 07 2007
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