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 A027907 Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n. 158
 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, 30, 45, 51, 45, 30, 15, 5, 1, 1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1, 1, 7, 28, 77, 161, 266, 357, 393, 357, 266, 161, 77, 28, 7, 1, 1, 8, 36, 112, 266 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS When the rows are centered about their midpoints, each term is the sum of the three terms directly above it (assuming the undefined terms in the previous row are zeros). - N. J. A. Sloane, Dec 23 2021 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_{i=0..floor(2*n/3)} T(n-i,i) = A000073(n+2). - Emeric Deutsch, 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, 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_2 X P_2 is: x*(x-1) - 2*x*(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 into each urn. - N-E. Fahssi, Mar 16 2008 T(n,k) is the number of compositions of k into n parts p, each part 0 <= p <= 2. Adding 1 to each part, as a corollary, T(n,k) is the number of compositions of n+k into n parts p where 1 <= p <= 3. E.g., T(2,3)=2 since 5 = 3+2 = 2+3. - Steffen Eger, Jun 10 2011 Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (1,2). - Joerg Arndt, Jul 05 2011 Number of lattice paths from (0,0) to (2*n-k,k) using steps (2,0), (1,1), (0,2). - Werner Schulte, Jan 25 2017 T(n,k) is number of distinct ways to sum the integers -1, 0 , and 1 n times to obtain n-k, where T(n,0) = T(n,2*n+1) = 1. - William Boyles, Apr 23 2017 T(n-1,k-1) is the number of 2-compositions of n with 0's having k parts; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 15 2020 REFERENCES 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. L. Carlitz, Comment on the paper "Some probability distributions and their associated structures", Math. Magazine, 37:1 (1964), 51-52. [The triangle is on page 51] 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). L. Kleinrock, Uniform permutation of sequences, JPL Space Programs Summary, Vol. 37-64-III, Apr 30, 1970, pp. 32-43. LINKS Seiichi Manyama, Rows n=0..99 of triangle, flattened (Rows 0..30 from T. D. Noe) Moussa Ahmia and Hacene Belbachir, Preserving log-convexity for generalized Pascal triangles, Electronic Journal of Combinatorics, 19(2) (2012), #P16. - N. J. A. Sloane, Oct 13 2012 Tewodros Amdeberhan, Moa Apagodu and Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015. Said Amrouche and Hacène Belbachir, Asymmetric extension of Pascal-Dellanoy triangles, arXiv:2001.11665 [math.CO], 2020. G. E. Andrews, Euler's 'exemplum memorabile inductionis fallacis' and q-trinomial coefficients, J. Amer. Math. Soc. 3 (1990) 653-669. G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p. George E. Andrews and Alexander Berkovich, A trinomial analogue of Bailey's lemma and N= 2 superconformal invariance, arXiv:q-alg/9702008, 1997; Communications in mathematical physics 192.2 (1998): 245-260. See page 248. Armen G. Bagdasaryan and Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77. Abdelghafour Bazeniar, Moussa Ahmia and Hacène Belbachir, Connection between bi^s nomial coefficients with their analogs and symmetric functions, Turkish Journal of Mathematics, Vol. 42, No. 3 (2018), pp. 807-818. Hacène Belbachir and Oussama Igueroufa, Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54. Hacène Belbachir and Yassine Otmani, Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences, Integers (2023) Vol. 23. Leonardo Bennun, A Pragmatic Smoothing Method for Improving the Quality of the Results in Atomic Spectroscopy, arXiv:1603.02061 [physics.atom-ph], 2016. See reference 22. Alexander Berkovich and Ali K. Uncu, Elementary Polynomial Identities Involving q-Trinomial Coefficients, arXiv:1810.06497 [math.NT], 2018. F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112. Jan Bok, Graph-indexed random walks on special classes of graphs, arXiv:1801.05498 [math.CO], 2018. Richard C. Bollinger, Reliability and Runs of Ones, Mathematics Magazine, 57(1) (1984), 34-37. 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. Eduardo H. M. Brietzke, Generalization of an identity of Andrews, Fibonacci Quart. 44(2) (2006), 166-171. Ji Young Choi, Digit Sums Generalizing Binomial Coefficients, J. Int. Seq. 22 (2019), Article 19.8.3. Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 7. S. Eger, Some Elementary Congruences for the Number of Weighted Integer Compositions, J. Int. Seq. 18 (2015), #15.4.1. L. Euler, Disquitiones analyticae super evolutione potestatis trinomialis (1+x+xx)^n, 1805. This is paper E722 in Eneström's index of Euler's works, translated from Latin to German. The sequence appears in the table on page 2. L. Euler, On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc.)^n, arXiv:math/0505425 [math.HO], 2005. L. Euler, De evolutione potestatis polynomialis cuiuscunque (1 + x + x^2 + x^3 + x^4 + etc.)^n, E709. Nour-Eddine Fahssi, Polynomial Triangles Revisited, arXiv:1202.0228 [math.CO], 2012. Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv:1203.6792 [math.CO], 2012  and J. Int. Seq., 17 (2014), #14.1.5. D. Fielder, Letter to N. J. A. Sloane, Jun. 1991. D. C. Fielder and C. O. Alford, Pascal's triangle: top gun or just one of the gang?, Applications of Fibonacci Numbers 4 (1991), 77-90. (Annotated scanned copy) S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008. A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math. 3(2) (2008), 76-114. W. Florek and T. Lulek, Combinatorial analysis of magnetic configurations, Séminaire Lotharingien de Combinatoire, B26d (1991), 12 pp. J. E. Freund, Restricted Occupancy Theory - A Generalization of Pascal's Triangle, American Mathematical Monthly, Vol. 63(1) (1956), 20-27. Berit Nilsen Givens, The trinomial triangle knitted shawl, J. Math. Arts (2023). R. K. Guy, Letter to N. J. A. Sloane, 1987 V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart. 7 (1969), 341-358 and 393. Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020. Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. S. Kak, The Golden Mean and the Physics of Aesthetics, arXiv:physics/0411195 [physics.hist-ph], 2004. L. Kleinrock, Uniform permutation of sequences, JPL Space Programs Summary, Vol. 37-64-III, Apr 30, 1970, pp. 32-43. [Annotated scanned copy] László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018. T. Neuschel, A Note on Extended Binomial Coefficients, J. Int. Seq. 17 (2014), #14.10.4. Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.] L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239. Eric Weisstein's World of Mathematics, Trinomial Triangle. Eric Weisstein's World of Mathematics, Trinomial Coefficient. Sheng-Liang Yang and Yuan-Yuan Gao, The Pascal rhombus and Riordan arrays, Fib. Q., 56:4 (2018), 337-347. See Fig. 3. Xuxu Zhao, Xu Wang and Haiyuan Yao, Some enumerative properties of a class of Fibonacci-like cubes, arXiv:1905.00573 [math.CO], 2019. Bao-Xuan Zhu, Linear transformations and strong q-log-concavity for certain combinatorial triangle, arXiv:1605.00257 [math.CO], 2016. FORMULA G.f.: 1/(1-z*(1+w+w^2)). T(n,k) = Sum_{r=0..floor(k/3)} (-1)^r*binomial(n, r)*binomial(k-3*r+n-1, n-1)). Recurrence: T(0,0) = 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k-0), with T(n,k) = 0 if k < 0 or k > 2*n: T(i,0) = T(i, 2*i) = 1 for i >= 0, T(i, 1) = T(i, 2*i-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, Aug 14 2004 T(n,k) = Sum_{i=0..floor(k/2)} binomial(n, 2*i+n-k) * binomial(2*i+n-k, i). - Ralf Stephan, Jan 26 2005 T(n,k) = Sum_{j=0..n} binomial(n, j) * binomial(j, k-j). - Paul Barry, May 21 2005 T(n,k) = Sum_{j=0..n} binomial(k-j, j) * binomial(n, k-j). - Paul Barry, Nov 04 2005 From Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006: (Start) T(n,k) = Sum_{j=0..n} (-1)^j * binomial(n,j) * binomial(2*n-2*j, k-j); (G. E. Andrews (1990)) obtained by expanding ((1+x)^2 - x)^n. T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(n-j, k-2*j); obtained by expanding ((1+x) + x^2)^n. T(n,k) = (-1)^k*Sum_{j=0..n} (-3)^j * binomial(n,j) * binomial(2*n-2*j, k-j); obtained by expanding ((1-x)^2 + 3*x)^n. T(n,k) = (1/2)^k * Sum_{j=0..n} 3^j * binomial(n,j) * binomial(2*n-2*j, k-2*j); obtained by expanding ((1+x/2)^2 + (3/4)*x^2)^n. T(n,k) = (2^k/4^n) * Sum_{j=0..n} 3^j * binomial(n,j) * binomial(2*n-2*j, k); obtained by expanding ((1/2+x)^2 + 3/4)^n using T(n,k) = T(2*n-k). (End) From Paul D. Hanna, Apr 18 2012: (Start) Let A(x) be the g.f. of the flattened sequence, then: G.f.: A(x) = Sum_{n>=0} x^(n^2) * (1+x+x^2)^n. G.f.: A(x) = Sum_{n>=0} x^n*(1+x+x^2)^n * Product_{k=1..n} (1 - (1+x+x^2) * x^(4*k-3)) / (1 - (1+x+x^2)*x^(4*k-1)). G.f.: A(x) = 1/(1 - x*(1+x+x^2)/(1 + x*(1-x^2)*(1+x+x^2)/(1 - x^5*(1+x+x^2)/(1 + x^3*(1-x^4)*(1+x+x^2)/(1 - x^9*(1+x+x^2)/(1 + x^5*(1-x^6)*(1+x+x^2)/(1 - x^13* (1+x+x^2)/(1 + x^7*(1-x^8)*(1+x+x^2)/(1 - ...))))))))), a continued fraction. (End) Triangle: G.f. = Sum_{n>=0} (1+x+x^2)^n * x^(n^2) * y^n. - Daniel Forgues, Mar 16 2015 From Peter Luschny, May 08 2016: (Start) T(n+1,n)/(n+1) = A001006(n) (Motzkin) for n>=0. T(n,k) = H(n, k) if k < n else H(n, 2*n-k) where H(n,k) = binomial(n,k)* hypergeom([(1-k)/2, -k/2], [n-k+1], 4)). T(n,k) = GegenbauerC(m, -n, -1/2) where m=k if k < n else 2*n-k. (End) T(n,k) = (-1)^k * C(2*n,k) * hypergeom([-k, -(2*n-k)], [-n+1/2], 3/4), for all k with 0 <= k <= 2n. - Robert S. Maier, Jun 13 2023 T(n,n) = Sum_{k=0..2*n} (-1)^k*(T(n,k))^2 and T(2*n,2*n) = Sum_{k=0..2*n} (T(n,k))^2 for n >= 0. - Werner Schulte, Nov 08 2016 T(n,n) = A002426(n), central trinomial coefficients. - M. F. Hasler, Nov 02 2019 Sum_{k=0..n-1} T(n, 2*k) = (3^n-1)/2. - Tony Foster III, Oct 06 2020 EXAMPLE The triangle T(n, k) begins: n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0: 1 1: 1 1 1 2: 1 2 3 2 1 3: 1 3 6 7 6 3 1 4: 1 4 10 16 19 16 10 4 1 5: 1 5 15 30 45 51 45 30 15 5 1 6: 1 6 21 50 90 126 141 126 90 50 21 6 1 Concatenated rows: G.f. = 1 + (x^2+x+1)*x + (x^2+x+1)^2*x^4 + (x^2+x+1)^3*x^9 + ... = 1 + (x + x^2 + x^3) + (x^4 + 2*x^5 + 3*x^6 + 2*x^7 + x^8) + (x^9 + 3*x^10 + 6*x^11 + 7*x^12 + 6*x^13 + 3*x^14 + x^15) + ... . As a centered triangle, this begins: ...........1........... ........1..1..1........ .....1..2..3..2..1..... ..1..3..6..7..6..3..1.. ...... MAPLE A027907 := proc(n, k) expand((1+x+x^2)^n) ; coeftayl(%, x=0, k) ; end proc: seq(seq(A027907(n, k), k=0..2*n), n=0..5) ; # R. J. Mathar, Jun 13 2011 T := (n, k) -> simplify(GegenbauerC(`if`(k

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