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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}.
3
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
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
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
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
Sequence in context: A103448 A186904 A216216 * A368227 A368179 A130403
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Dec 06 2002
EXTENSIONS
More terms from Emeric Deutsch, Jun 16 2007
STATUS
approved