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Search: id:A105422
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| A105422 |
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Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 1. |
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5
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| 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 3, 5, 3, 4, 0, 1, 5, 8, 9, 4, 5, 0, 1, 8, 15, 15, 14, 5, 6, 0, 1, 13, 26, 31, 24, 20, 6, 7, 0, 1, 21, 46, 57, 54, 35, 27, 7, 8, 0, 1, 34, 80, 108, 104, 85, 48, 35, 8, 9, 0, 1, 55, 139, 199, 209, 170, 125, 63, 44, 9, 10, 0, 1, 89, 240, 366, 404, 360
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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T(n,k) is also the number of length n bit strings beginning with 0 having k singletons. Example: T(4,2)=3 because we have 0010, 0100 and 0110. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 21 2008]
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REFERENCES
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D. Baccherini, D. Merlini and R. Sprugnoli, Level generating trees and proper Riordan arrays, Applicable Analysis and Discrete Mathematics, 2, 2008, 69-91 (see p. 83). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 21 2008]
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FORMULA
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G.f.=(1-z)/(1-z-z^2-tz+tz^2).
T(n,k)=T(n-1,k)+T(n-2,k)+T(n-1,k-1)-T(n-2,k-1), T(0,0)=1, T(1,0)=0. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2009]
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EXAMPLE
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T(6,2)=9 because we have (1,1,4),(1,4,1),(4,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1) and (2,2,1,1).
Triangle begins:
1;
0,1;
1,0,1;
1,2,0,1;
2,2,3,0,1;
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MAPLE
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G:=(1-z)/(1-z-z^2-t*z+t*z^2): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 13 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
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CROSSREFS
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Column 0 yields A000045 (the Fibonacci numbers). Column 1 yields A006367. Column 2 yields A105423. Row sums yield A011782.
Sequence in context: A029275 A058739 A128627 this_sequence A166291 A162986 A128584
Adjacent sequences: A105419 A105420 A105421 this_sequence A105423 A105424 A105425
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KEYWORD
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nonn,tabl,new
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 07 2005
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