%I A027907
%S A027907 1,1,1,1,1,2,3,2,1,1,3,6,7,6,3,1,1,4,10,16,19,16,10,4,1,1,5,15,
%T A027907 30,45,51,45,30,15,5,1,1,6,21,50,90,126,141,126,90,50,21,6,1,
%U A027907 1,7,28,77,161,266,357,393,357,266,161,77,28,7,1,1,8,36,112,266
%N A027907 Triangle of trinomial coefficients T(n,k) (n >= 0, 0<=k<=2n), read by 
               rows (n-th row is obtained by expanding (1+x+x^2)^n).
%C A027907 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.
%C A027907 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
%C A027907 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
%C A027907 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
%C A027907 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
%D A027907 F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 
               (1999) 73-112.
%D A027907 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), 
               FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published 
               by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; 
               see p. 17.
%D A027907 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%D A027907 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).
%D A027907 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring 
               at most thrice, in preparation.
%D A027907 V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal 
               triangles, Fib. Quart., 7 (1969), 341-358, 393.
%D A027907 L. Kleinrock, Uniform permutation of sequences, JPL Space Programs Summary, 
               Vol. 37-64-III, Apr 30, 1970, pp. 32-43.
%D A027907 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, 
               Discrete Applied Math., 34 (1991), 229-239.
%D A027907 G. E. Andrews, "Euler's `exemplum memorabile inductionis fallacis' and 
               q-trinomial coefficients", J. Amer. Math. Soc. 3 (1990) 653-669.
%D A027907 Freund, J. E., Restricted Occupancy Theory - A Generalization of Pascal's 
               Triangle, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 
               20-27.
%H A027907 T. D. Noe, <a href="b027907.txt">Rows n=0..30 of triangle, flattened</
               a>
%H A027907 S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">
               Odd Entries in Pascal's Trinomial Triangle</a> (arXiv:0802.2654)
%H A027907 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">
               Three aspects of partitions</a>
%H A027907 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505425">On the expansion 
               of the power of any polynomial (1+x+x^2+x^3+x^4+etc)^n</a>
%H A027907 L. Euler, <a href="http://www.eulerarchive.org">De evolutione potestatis 
               polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc)^n</a> E709
%H A027907 W. Florek and T. Lulek, <a href="http://www.mat.univie.ac.at/~slc/opapers/
               s26florek.html">Combinatorial analysis of magnetic configurations</
               a>
%H A027907 S. Kak, <a href="http://uk.arXiv.org/abs/physics/0411195">The Golden 
               Mean and the Physics of Aesthetics</a>
%H A027907 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TrinomialTriangle.html">Link to a section of The World of Mathematics.</
               a>
%H A027907 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TrinomialCoefficient.html">Trinomial Coefficient</a>
%F A027907 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).
%F A027907 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).
%F A027907 The row sums are powers of 3 (A000244). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Aug 14 2004
%F A027907 T(n, k) = Sum{i=0..[k/2], C(n, 2i+n-k)*C(2i+n-k, i) }. - R. Stephan, 
               Jan 26 2005
%F A027907 T(n, k):=sum{j=0..n, C(n, j)C(j, k-j)} - Paul Barry (pbarry(AT)wit.ie), 
               May 21 2005
%F A027907 T(n, k)=sum{j=0..n, C(k-j, j)C(n, k-j)}; - Paul Barry (pbarry(AT)wit.ie), 
               Nov 04 2005
%F A027907 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
%F A027907 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
%F A027907 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
%F A027907 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
%F A027907 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
%e A027907 1; 1,1,1; 1,2,3,2,1; 1,3,6,7,6,3,1; ...
%p A027907 # To get n-th row: expand((1+x+x^2)^n);
%o A027907 (PARI) T(n,k)=if(n<0,0,polcoeff((1+x+x^2)^n,k))
%Y A027907 Columns of T include A002426, A005717, A014531, A005581, A005712 etc. 
               See also A035000, A008287.
%Y A027907 Cf. A000073.
%Y A027907 First differences are in A025177. Pairwise sums are in A025564.
%Y A027907 Cf. A123531.
%Y A027907 Sequence in context: A092542 A026552 A086437 this_sequence A026323 A017838 
               A058294
%Y A027907 Adjacent sequences: A027904 A027905 A027906 this_sequence A027908 A027909 
               A027910
%K A027907 nonn,tabf,nice
%O A027907 0,6
%A A027907 N. J. A. Sloane (njas(AT)research.att.com).