0,6

T(n,k) = number of integer strings s(0),...,s(n) such that s(0)=0, s(n)=k, s(i)=s(i-1)+c, where c is 0, 1 or 2. Columns of T include A002426, A005717 and A014531.

Also number of ordered trees having n+1 leaves, all at level three and n+k+3 edges. Example: T(3,5)=3 because we have three ordered trees with 4 leaves, all at level three and 11 edges: the root r has three children; from one of these children two paths of length two are hanging (i.e. 3 possibilities) while from each of the other two children one path of length two is hanging. Diagonal sums are the tribonacci numbers; more precisely: Sum(T(n-i,i), i=0..floor(2n/3)) = A000073(n+2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 03 2004

T(n,k) = A111808(n,k) for 0<=k<=n and T(n,2*n-k) = A111808(n,k) for 0<=k<n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 17 2005

The trinomial coefficients, T(n,i), are the absolute value of the coefficients of the chromatic polynomial of P_2 X P_n factored with x(x-1)^i terms. Example: The chromatic polynomial of P_2xP_2 is: x(x-1) - 2x(x-1)^2 + x(x-1)^3 and so T(1,0)=1, T(1,1)=2 and T(1,1)=1. - Thomas J Pfaff (tpfaff(AT)ithaca.edu), Oct 02 2006

T(n,k) is the number of distinct ways in which k unlabeled objects can be distributed in n labeled urns allowing at most 2 objects to fall in each urn. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Mar 16 2008

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.

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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.

D. C. Fielder and C. O. Alford, Pascal's triangle: top gun or just one of the gang?, in G E Bergum et al., eds., Applications of Fibonacci Numbers Vol. 4 1991 pp. 77-90 (Kluwer).

A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.

V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.

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G. E. Andrews, "Euler's `exemplum memorabile inductionis fallacis' and q-trinomial coefficients", J. Amer. Math. Soc. 3 (1990) 653-669.

Freund, J. E., Restricted Occupancy Theory - A Generalization of Pascal's Triangle, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 20-27.

T. D. Noe, Rows n=0..30 of triangle, flattened

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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Trinomial Coefficient

G.f.: 1/(1-z(1+w+w^2)). T(n, k) = Sum_{0 <= r <= k/3} (-1)^r*C(n, r)*C(k-3*r+n-1, n-1).

T(i, 0) = T(i, 2i) = 1 for i >= 0, T(i, 1) = T(i, 2i-1) = i for i >= 1 and for i >= 2 and 2 <= j <= i-2, T(i, j) = T(i-1, j-2)+T(i-1, j-1)+T(i-1, j).

The row sums are powers of 3 (A000244). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 14 2004

T(n, k) = Sum{i=0..[k/2], C(n, 2i+n-k)*C(2i+n-k, i) }. - R. Stephan, Jan 26 2005

T(n, k):=sum{j=0..n, C(n, j)C(j, k-j)} - Paul Barry (pbarry(AT)wit.ie), May 21 2005

T(n, k)=sum{j=0..n, C(k-j, j)C(n, k-j)}; - Paul Barry (pbarry(AT)wit.ie), Nov 04 2005

T(n,k)=sum{j=0..n,(-1)^j*C(n,j)*C(2n-2j,k-j)};(G. E. Andrews (1990)); obtained by expanding [(1+x)^2-x]^n. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

T(n,k)=sum{j=0..n,C(n,j)*C(n-j,k-2j)}; obtained by expanding [(1+x)+x^2]^n. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

T(n,k)=(-1)^k*sum{j=0..n,(-3)^j*C(n,j)*C(2n-2j,k-j)} (a); obtained by expanding [(1-x)^2+3x]^n. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

T(n,k)=(1/2)^k*sum{j=0..n,3^j*C(n,j)*C(2n-2j,k-2j)} (b); obtained by expanding [(1+x/2)^2+(3/4)*x^2]^n. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

T(n,k)=(2^k/4^n)*sum{j=0..n,3^j*C(n,j)*C(2n-2j,k)} (c); obtained by expanding [(1/2+x)^2+3/4]^n; follows from (c) using T(n,k)=T(2n-k). - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

1; 1,1,1; 1,2,3,2,1; 1,3,6,7,6,3,1; ...

# To get n-th row: expand((1+x+x^2)^n);

(PARI) T(n, k)=if(n<0, 0, polcoeff((1+x+x^2)^n, k))

Columns of T include A002426, A005717, A014531, A005581, A005712 etc. See also A035000, A008287.

Cf. A000073.

First differences are in A025177. Pairwise sums are in A025564.

Cf. A123531.

Sequence in context: A092542 A026552 A086437 this_sequence A026323 A017838 A058294

Adjacent sequences: A027904 A027905 A027906 this_sequence A027908 A027909 A027910

nonn,tabf,nice

N. J. A. Sloane (njas(AT)research.att.com).