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A027907
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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).
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113
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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
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OFFSET
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0,6
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COMMENTS
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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, 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_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. - 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]
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REFERENCES
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Moussa Ahmia and Hacene Belbachir, Preserving log-convexity for generalized Pascal triangles, Electronic Journal of Combinatorics, 19(2) (2012), #P16; http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v19i2p16/pdf, 2012.- From N. J. A. Sloane, Oct 13 2012
G. E. Andrews, "Euler's `exemplum memorabile inductionis fallacis' and q-trinomial coefficients", J. Amer. Math. Soc. 3 (1990) 653-669.
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.
N.-E. Fahssi, The polynomial triangles revisited, Arxiv preprint arXiv:1202.0228, 2012
N.-E. Fahssi, A Systematic Study of Polynomial Triangles, Electronic Journal of Combinatorics, 2012. - From N. J. A. Sloane, Jan 04 2013
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, Arxiv preprint arXiv:1203.6792, 2012. - From N. J. A. Sloane, Oct 03 2012
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).
Freund, J. E., Restricted Occupancy Theory - A Generalization of Pascal's Triangle, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 20-27.
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
L. Kleinrock, Uniform permutation of sequences, JPL Space Programs Summary, Vol. 37-64-III, Apr 30, 1970, pp. 32-43.
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LINKS
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T. D. Noe, Rows n=0..30 of triangle, flattened
G. E. Andrews, Three aspects of partitions
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
L. Euler, On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc.)^n
L. Euler, De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc.)^n E709
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math. 3 (2) (2008).
W. Florek and T. Lulek, Combinatorial analysis of magnetic configurations
S. Kak, The Golden Mean and the Physics of Aesthetics
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
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FORMULA
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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, 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, May 21 2005
T(n, k)=sum{j=0..n, C(k-j, j)C(n, k-j)}; - Paul Barry, 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
Contribution 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)
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EXAMPLE
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Triangle begins
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; ...
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MAPLE
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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
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MATHEMATICA
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Table[CoefficientList[Series[(Sum[x^i, {i, 0, 2}])^n, {x, 0, 2 n}], x], {n, 0, 10}] // Grid [From Geoffrey Critzer, Mar 31 2010]
Table[Sum[Binomial[n, i]Binomial[n - i, k - 2i], {i, 0, n}], {n, 0, 10}, {k, 0, 2n}] [Adi Dani, May 07 2011]
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PROG
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(PARI) T(n, k)=if(n<0, 0, polcoeff((1+x+x^2)^n, k))
(Maxima) trinomial(n, k):=coeff(expand((1+x+x^2)^n), x, k);
create_list(trinomial(n, k), n, 0, 8, k, 0, 2*n); [Emanuele Munarini, Mar 15 2011]
(Haskell)
a027907 n k = a027907_tabf !! n !! k
a027907_row n = a027907_tabf !! n
a027907_tabf = iterate ([1, 1, 1] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
(p:ps) + (q:qs) = p + q : ps + qs
ps + qs = ps ++ qs
(p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
_ * _ = []
a027907_list = concat a027907_tabf
-- Reinhard Zumkeller, Jan 22 2013, Apr 02 2011
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CROSSREFS
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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.
Cf. A055217, A027914.
Sequence in context: A208233 A176270 A086437 * A026323 A017838 A181567
Adjacent sequences: A027904 A027905 A027906 * A027908 A027909 A027910
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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