login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008287 Triangle of quadrinomial coefficients, row n is the sequence of coefficients of (1 + x + x^2 + x^3)^n. 32
1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1, 1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Coefficient of x^k in (1 + x + x^2 + x^3)^n is the number of distinct ways in which k unlabeled objects can be distributed in n labeled urns allowing at most 3 objects to fall in each urn. - N-E. Fahssi, Mar 16 2008

Rows of A008287 mod 2 converted to decimal equals A177882. - Vladimir Shevelev, Jan 02 2011

T(n,k) is the number of compositions of k into n parts p, each part 0<=p<=3. 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<=4. E.g., T(2,3)=4 since 3=0+3=3+0=1+2=2+1. In general, the entry (n,k) of the (l+1)-nomial triangle gives the number of compositions of k into n parts p, each part 0<=p<=l. - Steffen Eger, Jun 18 2011

Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (1,2), (1,3). - Joerg Arndt, Jul 05 2011

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).

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..25 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

Spiros D. Dafnis, Frosso S. Makri, and Andreas N. Philippou, Restricted occupancy of s kinds of cells and generalized Pascal triangles, Fibonacci Quart. 45 (2007), no. 4, 347-356.

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).

J. E. Freund, Restricted Occupancy Theory - A Generalization of Pascal's Triangle, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 20-27.

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.

W. Florek and T. Lulek, Combinatorial analysis of magnetic configurations, Séminaire Lotharingien de Combinatoire, B26d (1991), 12 pp.

Claudia Smith and Verner E. Hoggatt, Jr. , A Study of the Maximal Values in Pascal's Quadrinomial Triangle, Fibonacci Quart. 17 (1979), no. 3, 264-269.

FORMULA

n-th row is formed by expanding (1+x+x^2+x^3)^n.

From Vladimir Shevelev, Dec 15 2010: (Start)

T(n,0) = 1; T(n,3*n) = 1; T(n,k) = T(n,3*n-k);

T(n,k) = 0, iff k<0 or k>3*n; Sum{k=0..3*n} T(n,k) = 4^n; Sum{k=0..3*n}((-1)^k)*T(n,k)=0 for n > 0; [corrected by Werner Schulte, Sep 09 2015]

T(n,k) = Sum{i=0..floor(k/2)} C(n,i)*C(n,k-2*i);

T(n+1,k) = T(n,k-3)+T(n,k-2)+T(n,k-1)+T(n,k). (End)

T(n,k) = sum {i = 0..floor(k/4)} (-1)^i*C(n,i)*C(n+k-1-4*i,n-1) for n >= 0 and 0 <= k <= 3*n. - Peter Bala, Sep 07 2013

G.f.: 1/(1 - ( x + y*x + y^2*x +y^3*x )). - Geoffrey Critzer, Feb 05 2014

T(n,k) = Sum_{j=0..k} (-2)^j*binomial(n,j)*binomial(3*n-2*j,k-j) for n >= 0 and 0 <= k <= 3*n (conjectured). - Werner Schulte, Sep 09 2015

EXAMPLE

Triangle begins

1;

1,1,1,1;

1,2,3,4,3,2,1;

1,3,6,10,12,12,10,6,3,1; ...

MAPLE

#Define the r-nomial coefficients for r = 1, 2, 3, ...

rnomial := (r, n, k) -> add((-1)^i*binomial(n, i)*binomial(n+k-1-r*i, n-1), i = 0..floor(k/r)):

#Display the 4-nomials as a table

r := 4:  rows := 10:

for n from 0 to rows do

seq(rnomial(r, n, k), k = 0..(r-1)*n)

end do;

# Peter Bala, Sep 07 2013

MATHEMATICA

Flatten[Table[CoefficientList[(1 + x + x^2 + x^3)^n, x], {n, 0, 10}]] (* T. D. Noe, Apr 04 2011 *)

PROG

(Maxima) quadrinomial(n, k):=coeff(expand((1+x+x^2+x^3)^n), x, k);

create_list(quadrinomial(n, k), n, 0, 8, k, 0, 3*n); /* Emanuele Munarini, Mar 15 2011 */

(Haskell)

a008287 n = a008287_list !! n

a008287_list = concat $ iterate ([1, 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

   _ * _               = []

-- Reinhard Zumkeller, Apr 02 2011

CROSSREFS

Cf. A007318, A027907, A177882.

Sequence in context: A017869 A107469 A167600 * A017859 A171456 A028356

Adjacent sequences:  A008284 A008285 A008286 * A008288 A008289 A008290

KEYWORD

nonn,tabf,easy,nice,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 13:17 EDT 2017. Contains 284270 sequences.