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 A027907 Irregular 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). 145
 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 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(2n/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) - 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 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,2n+1) = 1. - William Boyles, Apr 23 2017 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. 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. Sheng-Liang Yang et al., The Pascal rhombus and Riordan arrays, Fib. Q., 56:4 (2018), 337-347. See Fig. 3. 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, Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015. 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, Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77. Abdelghafour Bazeniar, Moussa Ahmia, Hacène Belbachir, Connection between bi^s nomial coefficients with their analogs and symmetric functions, Turkish Journal of Mathematics, August 2017. Leonardo Bennun, A Pragmatic Smoothing Method for Improving the Quality of the Results in Atomic Spectroscopy, arXiv preprint arXiv:1603.02061 [physics.atom-ph], 2016. See reference 22. Alexander Berkovich, 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. 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 (2006), no. 2, 166-171. 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. Nour-Eddine Fahssi, Polynomial Triangles Revisited, arXiv:1202.0228 [math.CO], (25-July-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), pp. 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, No. 1 (1956), pp. 20-27. 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, 393. 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. 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 Xuxu Zhao, Xu Wang, 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 preprint 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 > 2n: 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, Aug 14 2004 T(n,k) = Sum_{i=0..floor(k/2)} binomial(n, 2i+n-k) * binomial(2i+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(2n-2j, 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-2j); obtained by expanding ((1+x) + x^2)^n. T(n,k) = (-1)^k*Sum_{j=0..n} (-3)^j * binomial(n,j) * binomial(2n-2j, k-j); obtained by expanding ((1-x)^2 + 3x)^n. T(n,k) = (1/2)^k * Sum_{j=0..n} 3^j * binomial(n,j) * binomial(2n-2j, k-2j); 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(2n-2j, k); obtained by expanding ((1/2+x)^2 + 3/4)^n using T(n,k) = T(2n-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,n) = Sum_{k=0..2n} (-1)^k*(T(n,k))^2 and T(2n,2n) = Sum_{k=0..2n} (T(n,k))^2 for n >= 0. - Werner Schulte, Nov 08 2016 EXAMPLE The irregular 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 ... reformatted - Wolfdieter Lang, Oct 31 2015 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) + ... . 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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Last modified October 22 19:53 EDT 2019. Contains 328319 sequences. (Running on oeis4.)