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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}.

%I A078803 #45 May 28 2025 04:53:37
%S A078803 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 A078803 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 A078803 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 A078803 Triangular array T given by T(n,k) = number of compositions of n into k parts, each in the set {1,2,3}.
%C A078803 Number of lattice paths from (0,0) to (n,k) using steps (1,1), (2,1), (3,1). - _Joerg Arndt_, Jul 05 2011
%C A078803 Reversing the rows produces A078802. Row sums: A000073.
%C A078803 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 A078803 This is the Riordan array (1,x+x^2+x^3)(A071675) without its column k=0. - _Vladimir Kruchinin_, Feb 10 2011
%D A078803 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 A078803 Alois P. Heinz, <a href="/A078803/b078803.txt">Rows n = 1..141, flattened</a>
%H A078803 Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, <a href="http://jl.baril.u-bourgogne.fr/narayana.pdf">Generalized Narayana arrays, restricted Dyck paths, and related bijections</a>, Univ. Bourgogne (France, 2025). See p. 18.
%H A078803 Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%F A078803 T(n, k) = t(n-1, n-k), for 1<=k<=n, for n>=1, where the array t is given by A078802.
%F A078803 G.f.: 1/(1-t*z*(1+z+z^2))-1. - _Emeric Deutsch_, Mar 10 2004
%F A078803 T(n,k) = Sum_{j=0..k} C(j,n-3*k+2*j)*C(k,j). - _Vladimir Kruchinin_, Feb 10 2011
%e A078803 T(5,2) = 2 counts the compositions 2+3 and 3+2.
%e A078803 Triangle begins
%e A078803   1;
%e A078803   1, 1;
%e A078803   1, 2, 1;
%e A078803   0, 3, 3, 1;
%e A078803   0, 2, 6, 4, 1;
%e A078803   0, 1, 7, 10, 5, 1;
%e A078803   0, 0, 6, 16, 15, 6, 1;
%e A078803   0, 0, 3, 19, 30, 21, 7, 1;
%e A078803   0, 0, 1, 16, 45, 50, 28, 8, 1;
%e A078803   0, 0, 0, 10, 51, 90, 77, 36, 9, 1;
%e A078803   0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1;
%e A078803   0, 0, 0, 1, 30, 141, 266, 266, 156, 55, 11, 1;
%p A078803 A078803 := proc(n,k) add( binomial(j,n-3*k+2*j)*binomial(k,j),j=0..k) ; end proc:
%p A078803 # _R. J. Mathar_, Feb 22 2011
%t A078803 nn=8;CoefficientList[Series[1/(1-y(x+x^2+x^3)),{x,0,nn}],{x,y}]//Grid (* _Geoffrey Critzer_, Jan 08 2013 *)
%Y A078803 Cf. A027907, A078802, A030528 (parts <=2), A213887 (parts <=4), A213888 (parts <=5), A061676 and A213889 (parts <=6).
%K A078803 nonn,tabl
%O A078803 1,5
%A A078803 _Clark Kimberling_, Dec 06 2002
%E A078803 More terms from _Emeric Deutsch_, Jun 16 2007