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A105422 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 1. 6
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; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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. - Emeric Deutsch, Sep 21 2008

REFERENCES

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). - Emeric Deutsch, Sep 21 2008

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

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. - Philippe Deléham, Nov 18 2009

EXAMPLE

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:

00:  1,

01:  0, 1,

02:  1, 0, 1,

03:  1, 2, 0, 1,

04:  2, 2, 3, 0, 1,

05:  3, 5, 3, 4, 0, 1,

06:  5, 8, 9, 4, 5, 0, 1,

07:  8, 15, 15, 14, 5, 6, 0, 1,

08:  13, 26, 31, 24, 20, 6, 7, 0, 1,

09:  21, 46, 57, 54, 35, 27, 7, 8, 0, 1,

10:  34, 80, 108, 104, 85, 48, 35, 8, 9, 0, 1,

11:  55, 139, 199, 209, 170, 125, 63, 44, 9, 10, 0, 1,

12:  89, 240, 366, 404, 360, 258, 175, 80, 54, 10, 11, 0, 1,

13:  144, 413, 666, 780, 725, 573, 371, 236, 99, 65, 11, 12, 0, 1,

...

MAPLE

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

# second Maple program

T:= proc(n) option remember; local j; if n=0 then 1

      else []; for j to n do zip((x, y)-> x+y, %,

      [`if`(j=1, 0, [][]), T(n-j)], 0) od; %[] fi

    end:

seq(T(n), n=0..20);  # Alois P. Heinz, Nov 05 2012

MATHEMATICA

nn = 10; a = x/(1 - x) - x + y x;

CoefficientList[CoefficientList[Series[1/(1 - a), {x, 0, nn}], x], y] // Flatten   (* Geoffrey Critzer, Dec 23 2011 *)

CROSSREFS

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 * A166291 A162986 A128584

Adjacent sequences:  A105419 A105420 A105421 * A105423 A105424 A105425

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Apr 07 2005

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.