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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). 135
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(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 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

REFERENCES

Leonardo Bennun, A Pragmatic Smoothing Method for Improving the Quality of the Results in Atomic Spectroscopy, arXiv preprint arXiv:1603.02061, 2016. See reference 22.

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.

Bao-Xuan Zhu, Linear transformations and strong $ q $-log-concavity for certain combinatorial triangle, arXiv preprint arXiv:1605.00257, 2016

LINKS

T. D. Noe, Rows n=0..30 of triangle, flattened

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.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.

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.

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], (25-July-2012)

Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv:1203.6792 [math.CO], 2012.

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).

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.

V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.

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]

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

FORMULA

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)).

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..[k/2], C(n, 2i+n-k) * C(2i+n-k, i)}. - Ralf 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

From Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006: (Start)

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.

T(n,k) = Sum{j=0..n, C(n,j) * C(n-j, k-2j)}; obtained by expanding [(1+x)+x^2]^n.

T(n,k) = (-1)^k*Sum{j=0..n, (-3)^j * C(n,j) * C(2n-2j, k-j)}; obtained by expanding [(1-x)^2+3x]^n.

T(n,k) = (1/2)^k * Sum_{j=0..n, 3^j * C(n,j) * C(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 * C(n,j) * C(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<n, k, 2*n-k), -n, -1/2));

for n from 0 to 8 do seq(T(n, k), k=0..2*n) od; # Peter Luschny, May 08 2016

MATHEMATICA

Table[CoefficientList[Series[(Sum[x^i, {i, 0, 2}])^n, {x, 0, 2 n}], x], {n, 0, 10}] // Grid (* 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 *)

T[ n_, k_] := If[ n < 0, 0, Coefficient[ (1 + x + x^2)^n, x, k]]; (* Michael Somos, Nov 08 2016 *)

PROG

(PARI) {T(n, k) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, k))}; /* Michael Somos, Jun 27 2003 */

(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

(Maxima) create_list(ultraspherical(k, -n, -1/2), n, 0, 6, k, 0, 2*n); /* Emanuele Munarini, Oct 18 2016 */

(Haskell)

a027907 n k = a027907_tabf !! n !! k

a027907_row n = a027907_tabf !! n

a027907_tabf = [1] : iterate f [1, 1, 1] where

   f row = zipWith3 (((+) .) . (+))

                    (row ++ [0, 0]) ([0] ++ row ++ [0]) ([0, 0] ++ row)

a027907_list = concat a027907_tabf

-- Reinhard Zumkeller, Jul 06 2014, Jan 22 2013, Apr 02 2011

CROSSREFS

Columns of T include A002426, A005717, A014531, A005581, A005712, etc. See also A035000, A008287.

First differences are in A025177. Pairwise sums are in A025564.

Cf. A000073, A001006, A123531, A055217, A027908, A027913, A027914, A027023.

Sequence in context: A208233 A176270 A086437 * A026323 A017838 A181567

Adjacent sequences:  A027904 A027905 A027906 * A027908 A027909 A027910

KEYWORD

nonn,tabf,nice,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 26 10:42 EDT 2017. Contains 284111 sequences.