This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027907 #352 May 02 2025 10:30:03
%S A027907 1,1,1,1,1,2,3,2,1,1,3,6,7,6,3,1,1,4,10,16,19,16,10,4,1,1,5,15,30,45,
%T A027907 51,45,30,15,5,1,1,6,21,50,90,126,141,126,90,50,21,6,1,1,7,28,77,161,
%U A027907 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 <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n.
%C A027907 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
%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_{i=0..floor(2*n/3)} T(n-i,i) = A000073(n+2). - _Emeric Deutsch_, 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_, 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_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
%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 into each urn. - _N-E. Fahssi_, Mar 16 2008
%C A027907 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
%C A027907 Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (1,2). - _Joerg Arndt_, Jul 05 2011
%C A027907 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
%C A027907 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
%C A027907 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
%C A027907 T(n,k) is the number of ways to obtain a sum of n+k when throwing a 3-sided die n times. Follows from the "T(n,k) is the number of compositions of n+k into n parts p where 1 <= p <= 3" comment above. - _Feryal Alayont_, Dec 30 2024
%D A027907 Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%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 L. Kleinrock, Uniform permutation of sequences, JPL Space Programs Summary, Vol. 37-64-III, Apr 30, 1970, pp. 32-43.
%H A027907 Seiichi Manyama, <a href="/A027907/b027907.txt">Rows n=0..99 of triangle, flattened</a> (Rows 0..30 from T. D. Noe)
%H A027907 Moussa Ahmia and Hacene Belbachir, <a href="https://doi.org/10.37236/2255">Preserving log-convexity for generalized Pascal triangles</a>, Electronic Journal of Combinatorics, 19(2) (2012), #P16. - _N. J. A. Sloane_, Oct 13 2012
%H A027907 Tewodros Amdeberhan, Moa Apagodu and Doron Zeilberger, <a href="http://arxiv.org/abs/1507.07660">Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle</a>, arXiv:1507.07660 [math.CO], 2015.
%H A027907 Said Amrouche and Hacène Belbachir, <a href="https://arxiv.org/abs/2001.11665">Asymmetric extension of Pascal-Dellanoy triangles</a>, arXiv:2001.11665 [math.CO], 2020.
%H A027907 G. E. Andrews, <a href="http://dx.doi.org/10.1090/S0894-0347-1990-1040390-4">Euler's 'exemplum memorabile inductionis fallacis' and q-trinomial coefficients</a>, J. Amer. Math. Soc. 3 (1990) 653-669.
%H A027907 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects of partitions</a>, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
%H A027907 George E. Andrews and Alexander Berkovich, <a href="https://arxiv.org/abs/q-alg/9702008">A trinomial analogue of Bailey's lemma and N= 2 superconformal invariance</a>, arXiv:q-alg/9702008, 1997; Communications in mathematical physics 192.2 (1998): 245-260. See page 248.
%H A027907 Armen G. Bagdasaryan and Ovidiu Bagdasar, <a href="https://doi.org/10.1016/j.endm.2018.05.012">On some results concerning generalized arithmetic triangles</a>, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.
%H A027907 Abdelghafour Bazeniar, Moussa Ahmia and Hacène Belbachir, <a href="https://doi.org/10.3906/mat-1705-27">Connection between bi^s nomial coefficients with their analogs and symmetric functions</a>, Turkish Journal of Mathematics, Vol. 42, No. 3 (2018), pp. 807-818.
%H A027907 Hacène Belbachir and Oussama Igueroufa, <a href="https://hal.archives-ouvertes.fr/hal-02918958/document#page=48">Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers</a>, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54.
%H A027907 Hacène Belbachir and Yassine Otmani, <a href="http://math.colgate.edu/~integers/x27/x27.pdf">Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences</a>, Integers (2023) Vol. 23.
%H A027907 Leonardo Bennun, <a href="http://arxiv.org/abs/1603.02061">A Pragmatic Smoothing Method for Improving the Quality of the Results in Atomic Spectroscopy</a>, arXiv:1603.02061 [physics.atom-ph], 2016. See reference 22.
%H A027907 Alexander Berkovich and Ali K. Uncu, <a href="https://arxiv.org/abs/1810.06497">Elementary Polynomial Identities Involving q-Trinomial Coefficients</a>, arXiv:1810.06497 [math.NT], 2018.
%H A027907 F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999) 73-112.
%H A027907 Jan Bok, <a href="https://arxiv.org/abs/1801.05498">Graph-indexed random walks on special classes of graphs</a>, arXiv:1801.05498 [math.CO], 2018.
%H A027907 Richard C. Bollinger, <a href="https://www.jstor.org/stable/2690294">Reliability and Runs of Ones</a>, Mathematics Magazine, 57(1) (1984), 34-37.
%H A027907 Boris A. Bondarenko, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a>, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 17.
%H A027907 Eduardo H. M. Brietzke, <a href="https://www.fq.math.ca/Papers1/44-2/quarteduardobrietzke02_2006.pdf">Generalization of an identity of Andrews</a>, Fibonacci Quart. 44(2) (2006), 166-171.
%H A027907 Rezig Boualam and Moussa Ahmia, <a href="https://arxiv.org/abs/2409.18886">Log-concavity and strong q-log-convexity for some generalized triangular arrays</a>, arXiv:2409.18886 [math.CO], 2024. See p. 2.
%H A027907 L. Carlitz, <a href="https://www.jstor.org/stable/2688252">Comment on the paper "Some probability distributions and their associated structures"</a>, Math. Magazine, 37:1 (1964), 51-52. [The triangle is on page 51]
%H A027907 Ji Young Choi, <a href="https://www.emis.de/journals/JIS/VOL22/Choi/choi15.html">Digit Sums Generalizing Binomial Coefficients</a>, J. Int. Seq. 22 (2019), Article 19.8.3.
%H A027907 Karl Dilcher and Larry Ericksen, <a href="https://arxiv.org/abs/2405.12024">Polynomials and algebraic curves related to certain binary and b-ary overpartitions</a>, arXiv:2405.12024 [math.CO], 2024. See p. 7.
%H A027907 S. Eger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Eger/eger11.html">Some Elementary Congruences for the Number of Weighted Integer Compositions</a>, J. Int. Seq. 18 (2015), #15.4.1.
%H A027907 L. Euler, <a href="https://arxiv.org/abs/1202.0028">Disquitiones analyticae super evolutione potestatis trinomialis (1+x+xx)^n</a>, 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.
%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>, arXiv:math/0505425 [math.HO], 2005.
%H A027907 L. Euler, <a href="http://eulerarchive.maa.org/pages/E709.html">De evolutione potestatis polynomialis cuiuscunque (1 + x + x^2 + x^3 + x^4 + etc.)^n</a>, E709.
%H A027907 Nour-Eddine Fahssi, <a href="http://arxiv.org/abs/1202.0228">Polynomial Triangles Revisited</a>, arXiv:1202.0228 [math.CO], 2012.
%H A027907 Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv:1203.6792 [math.CO], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">J. Int. Seq., 17 (2014), #14.1.5</a>.
%H A027907 D. Fielder, <a href="/A027907/a027907_1.pdf">Letter to N. J. A. Sloane, Jun. 1991</a>.
%H A027907 D. C. Fielder and C. O. Alford, <a href="/A027907/a027907_2.pdf">Pascal's triangle: top gun or just one of the gang?</a>, Applications of Fibonacci Numbers 4 (1991), 77-90. (Annotated scanned copy)
%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 [math.NT], 2008.
%H A027907 A. Fink, R. K. Guy and M. Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contrib. Discr. Math. 3(2) (2008), 76-114.
%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>, Séminaire Lotharingien de Combinatoire, B26d (1991), 12 pp.
%H A027907 J. E. Freund, <a href="http://www.jstor.org/stable/2308048">Restricted Occupancy Theory - A Generalization of Pascal's Triangle</a>, American Mathematical Monthly, Vol. 63(1) (1956), 20-27.
%H A027907 Berit Nilsen Givens, <a href="https://doi.org/10.1080/17513472.2023.2197832">The trinomial triangle knitted shawl</a>, J. Math. Arts (2023).
%H A027907 Fern Gossow, <a href="https://arxiv.org/abs/2410.05678">Lyndon-like cyclic sieving and Gauss congruence</a>, arXiv:2410.05678 [math.CO], 2024. See p. 17.
%H A027907 R. K. Guy, <a href="/A005712/a005712.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A027907 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://www.fq.math.ca/Scanned/7-4/hoggatt-a.pdf">Diagonal sums of generalized Pascal triangles</a>, Fib. Quart. 7 (1969), 341-358 and 393.
%H A027907 Brian Hopkins and Stéphane Ouvry, <a href="https://arxiv.org/abs/2008.04937">Combinatorics of Multicompositions</a>, arXiv:2008.04937 [math.CO], 2020.
%H A027907 Veronika Irvine, <a href="http://hdl.handle.net/1828/7495">Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns</a>, PhD Dissertation, University of Victoria, 2016.
%H A027907 S. Kak, <a href="http://arxiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>, arXiv:physics/0411195 [physics.hist-ph], 2004.
%H A027907 L. Kleinrock, <a href="/A027907/a027907.pdf">Uniform permutation of sequences</a>, JPL Space Programs Summary, Vol. 37-64-III, Apr 30, 1970, pp. 32-43. [Annotated scanned copy]
%H A027907 László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%H A027907 T. Neuschel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Neuschel/neuschel4.html">A Note on Extended Binomial Coefficients</a>, J. Int. Seq. 17 (2014), #14.10.4.
%H A027907 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 4.
%H A027907 Andrei G. Pronko, <a href="https://arxiv.org/abs/2504.00835">Periodic Motzkin chain: Ground states and symmetries</a>, arXiv:2504.00835 [math-ph], 2025. See p. 16.
%H A027907 Jack Ramsay, <a href="/A349812/a349812.pdf">On Arithmetical Triangles</a>, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]
%H A027907 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan group</a>, Discrete Applied Math., 34 (1991), 229-239.
%H A027907 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialTriangle.html">Trinomial Triangle</a>.
%H A027907 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>.
%H A027907 Sheng-Liang Yang and Yuan-Yuan Gao, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/56-4/yanggao1032018.pdf">The Pascal rhombus and Riordan arrays</a>, Fib. Q., 56:4 (2018), 337-347. See Fig. 3.
%H A027907 Xuxu Zhao, Xu Wang and Haiyuan Yao, <a href="https://arxiv.org/abs/1905.00573">Some enumerative properties of a class of Fibonacci-like cubes</a>, arXiv:1905.00573 [math.CO], 2019.
%H A027907 Bao-Xuan Zhu, <a href="http://arxiv.org/abs/1605.00257">Linear transformations and strong q-log-concavity for certain combinatorial triangle</a>, arXiv:1605.00257 [math.CO], 2016.
%F A027907 G.f.: 1/(1-z*(1+w+w^2)).
%F A027907 T(n,k) = Sum_{r=0..floor(k/3)} (-1)^r*binomial(n, r)*binomial(k-3*r+n-1, n-1).
%F A027907 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:
%F A027907 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).
%F A027907 The row sums are powers of 3 (A000244). - _Gerald McGarvey_, Aug 14 2004
%F A027907 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
%F A027907 T(n,k) = Sum_{j=0..n} binomial(n, j) * binomial(j, k-j). - _Paul Barry_, May 21 2005
%F A027907 T(n,k) = Sum_{j=0..n} binomial(k-j, j) * binomial(n, k-j). - _Paul Barry_, Nov 04 2005
%F A027907 From Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006: (Start)
%F A027907 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.
%F A027907 T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(n-j, k-2*j); obtained by expanding ((1+x) + x^2)^n.
%F A027907 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.
%F A027907 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.
%F A027907 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)
%F A027907 From _Paul D. Hanna_, Apr 18 2012: (Start)
%F A027907 Let A(x) be the g.f. of the flattened sequence, then:
%F A027907 G.f.: A(x) = Sum_{n>=0} x^(n^2) * (1+x+x^2)^n.
%F A027907 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)).
%F A027907 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.
%F A027907 (End)
%F A027907 Triangle: G.f. = Sum_{n>=0} (1+x+x^2)^n * x^(n^2) * y^n. - _Daniel Forgues_, Mar 16 2015
%F A027907 From _Peter Luschny_, May 08 2016: (Start)
%F A027907 T(n+1,n)/(n+1) = A001006(n) (Motzkin) for n>=0.
%F A027907 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).
%F A027907 T(n,k) = GegenbauerC(m, -n, -1/2) where m=k if k < n else 2*n-k. (End)
%F A027907 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
%F A027907 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
%F A027907 T(n,n) = A002426(n), central trinomial coefficients. - _M. F. Hasler_, Nov 02 2019
%F A027907 Sum_{k=0..n-1} T(n, 2*k) = (3^n-1)/2. - _Tony Foster III_, Oct 06 2020
%e A027907 The triangle T(n, k) begins:
%e A027907 n\k 0 1 2 3 4 5 6 7 8 9 10 11 12
%e A027907 0: 1
%e A027907 1: 1 1 1
%e A027907 2: 1 2 3 2 1
%e A027907 3: 1 3 6 7 6 3 1
%e A027907 4: 1 4 10 16 19 16 10 4 1
%e A027907 5: 1 5 15 30 45 51 45 30 15 5 1
%e A027907 6: 1 6 21 50 90 126 141 126 90 50 21 6 1
%e A027907 Concatenated rows:
%e A027907 G.f. = 1 + (x^2+x+1)*x + (x^2+x+1)^2*x^4 + (x^2+x+1)^3*x^9 + ...
%e A027907 = 1 + (x + x^2 + x^3) + (x^4 + 2*x^5 + 3*x^6 + 2*x^7 + x^8) +
%e A027907 (x^9 + 3*x^10 + 6*x^11 + 7*x^12 + 6*x^13 + 3*x^14 + x^15) + ... .
%e A027907 As a centered triangle, this begins:
%e A027907 1
%e A027907 1 1 1
%e A027907 1 2 3 2 1
%e A027907 1 3 6 7 6 3 1
%p A027907 A027907 := proc(n,k) expand((1+x+x^2)^n) ; coeftayl(%,x=0,k) ; end proc:
%p A027907 seq(seq(A027907(n,k),k=0..2*n),n=0..5) ; # _R. J. Mathar_, Jun 13 2011
%p A027907 T := (n,k) -> simplify(GegenbauerC(`if`(k<n,k,2*n-k), -n, -1/2));
%p A027907 for n from 0 to 8 do seq(T(n,k),k=0..2*n) od; # _Peter Luschny_, May 08 2016
%t A027907 Table[CoefficientList[Series[(Sum[x^i, {i, 0, 2}])^n, {x, 0, 2 n}], x], {n, 0, 10}] // Grid (* _Geoffrey Critzer_, Mar 31 2010 *)
%t A027907 Table[Sum[Binomial[n, i]Binomial[n - i, k - 2i], {i, 0, n}], {n, 0, 10}, {k, 0, 2n}] (* _Adi Dani_, May 07 2011 *)
%t A027907 T[ n_, k_] := If[ n < 0, 0, Coefficient[ (1 + x + x^2)^n, x, k]]; (* _Michael Somos_, Nov 08 2016 *)
%t A027907 Flatten[DeleteCases[#,0]&/@CellularAutomaton[{Total[#] &, {}, 1}, {{1}, 0}, 8] ] (* _Giorgos Kalogeropoulos_, Nov 09 2021 *)
%o A027907 (PARI) {T(n, k) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, k))}; /* _Michael Somos_, Jun 27 2003 */
%o A027907 (Maxima) trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
%o A027907 create_list(trinomial(n,k),n,0,8,k,0,2*n); /* _Emanuele Munarini_, Mar 15 2011 */
%o A027907 (Maxima) create_list(ultraspherical(k,-n,-1/2),n,0,6,k,0,2*n); /* _Emanuele Munarini_, Oct 18 2016 */
%o A027907 (Haskell)
%o A027907 a027907 n k = a027907_tabf !! n !! k
%o A027907 a027907_row n = a027907_tabf !! n
%o A027907 a027907_tabf = [1] : iterate f [1, 1, 1] where
%o A027907 f row = zipWith3 (((+) .) . (+))
%o A027907 (row ++ [0, 0]) ([0] ++ row ++ [0]) ([0, 0] ++ row)
%o A027907 a027907_list = concat a027907_tabf
%o A027907 -- _Reinhard Zumkeller_, Jul 06 2014, Jan 22 2013, Apr 02 2011
%Y A027907 Columns of T include A002426, A005717, A014531, A005581, A005712, etc. See also A035000, A008287.
%Y A027907 First differences are in A025177. Pairwise sums are in A025564.
%Y A027907 Cf. A007318 (Pascal's triangle); A000073, A001006, A123531, A055217, A027908, A027913, A027914, A027023.
%K A027907 nonn,tabf,nice,easy
%O A027907 0,6
%A A027907 _N. J. A. Sloane_