%I #36 Aug 22 2024 20:52:51
%S 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,
%T 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,
%U 126,161,112,45,10,1,0,0,0,1,30,141,266,266,156,55,11,1,0,0,0,0,15,126
%N Triangular array T given by T(n,k) = number of compositions of n into k parts, each in the set {1,2,3}.
%C Number of lattice paths from (0,0) to (n,k) using steps (1,1), (2,1), (3,1). - _Joerg Arndt_, Jul 05 2011
%C Reversing the rows produces A078802. Row sums: A000073.
%C 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
%C This is the Riordan array (1,x+x^2+x^3)(A071675) without its column k=0. - _Vladimir Kruchinin_, Feb 10 2011
%D 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.
%H Alois P. Heinz, <a href="/A078803/b078803.txt">Rows n = 1..141, flattened</a>
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%F T(n, k)=t(n-1, n-k), for 1<=k<=n, for n>=1, where the array t is given by A078802.
%F G.f.: 1/(1-t*z*(1+z+z^2))-1. - _Emeric Deutsch_, Mar 10 2004
%F T(n,k) = Sum_{j=0..k} C(j,n-3*k+2*j)*C(k,j). - _Vladimir Kruchinin_, Feb 10 2011
%e T(5,2) = 2 counts the compositions 2+3 and 3+2.
%e Triangle begins
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 0, 3, 3, 1;
%e 0, 2, 6, 4, 1;
%e 0, 1, 7, 10, 5, 1;
%e 0, 0, 6, 16, 15, 6, 1;
%e 0, 0, 3, 19, 30, 21, 7, 1;
%e 0, 0, 1, 16, 45, 50, 28, 8, 1;
%e 0, 0, 0, 10, 51, 90, 77, 36, 9, 1;
%e 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1;
%e 0, 0, 0, 1, 30, 141, 266, 266, 156, 55, 11, 1;
%p A078803 := proc(n,k) add( binomial(j,n-3*k+2*j)*binomial(k,j),j=0..k) ; end proc:
%p # _R. J. Mathar_, Feb 22 2011
%t nn=8;CoefficientList[Series[1/(1-y(x+x^2+x^3)),{x,0,nn}],{x,y}]//Grid (* _Geoffrey Critzer_, Jan 08 2013 *)
%Y Cf. A027907, A078802.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Dec 06 2002
%E More terms from _Emeric Deutsch_, Jun 16 2007