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A078803 Triangular array T given by T(n,k) = number of compositions of n into k parts, each in the set {1,2,3}. 2
1, 1, 1, 1, 2, 1, 0, 3, 3, 1, 0, 2, 6, 4, 1, 0, 1, 7, 10, 5, 1, 0, 0, 6, 16, 15, 6, 1, 0, 0, 3, 19, 30, 21, 7, 1, 0, 0, 1, 16, 45, 50, 28, 8, 1, 0, 0, 0, 10, 51, 90, 77, 36, 9, 1, 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1, 0, 0, 0, 1, 30, 141, 266, 266, 156, 55, 11, 1, 0, 0, 0, 0, 15, 126 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Number of lattice paths from (0,0) to (n,k) using steps (1,1), (2,1), (3,1). - Joerg Arndt, Jul 05 2011

Reversing the rows produces A078802. Row sums: A000073.

Number of tribonacci binary words of length n-1 having k-1 1's. A tribonacci binary word is a binary word having no three consecutive 0's. Example: T(6,3)=7 because we have 00101,00110,01001,01010,01100,10010 and 10100. - Emeric Deutsch, Jun 16 2007

This is the Riordan array (1,x+x^2+x^3)(A071675) without its column k=0. - Vladimir Kruchinin, Feb 10 2011

REFERENCES

Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565

FORMULA

T(n, k)=t(n-1, n-k), for 1<=k<=n, for n>=1, where the array t is given by A078802.

G.f.: 1/(1-t*z*(1+z+z^2))-1. - Emeric Deutsch, Mar 10 2004

T(n,k) = Sum_{j=0..k} C(j,n-3*k+2*j)*C(k,j). - Vladimir Kruchinin, Feb 10 2011]

EXAMPLE

T(5,2) = 2 counts the compositions 2+3 and 3+2.

Triangle begins

1;

1, 1;

1, 2, 1;

0, 3, 3, 1;

0, 2, 6, 4, 1;

0, 1, 7, 10, 5, 1;

0, 0, 6, 16, 15, 6, 1;

0, 0, 3, 19, 30, 21, 7, 1;

0, 0, 1, 16, 45, 50, 28, 8, 1;

0, 0, 0, 10, 51, 90, 77, 36, 9, 1;

0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1;

0, 0, 0, 1, 30, 141, 266, 266, 156, 55, 11, 1;

MAPLE

A078803 := proc(n, k) add( binomial(j, n-3*k+2*j)*binomial(k, j), j=0..k) ; end proc:

# R. J. Mathar, Feb 22 2011

MATHEMATICA

nn=8; CoefficientList[Series[1/(1-y(x+x^2+x^3)), {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, Jan 08 2013 *)

CROSSREFS

Cf. A027907, A078802.

Sequence in context: A103448 A186904 A216216 * A130403 A130402 A089840

Adjacent sequences:  A078800 A078801 A078802 * A078804 A078805 A078806

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Dec 06 2002

EXTENSIONS

More terms from Emeric Deutsch, Jun 16 2007

STATUS

approved

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Last modified November 17 06:06 EST 2019. Contains 329217 sequences. (Running on oeis4.)