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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, 456, 336, 216, 120, 56, 21, 6, 1 (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).

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).

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)

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.

R. K. Guy, Letter to N. J. A. Sloane, 1987

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.

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

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 *)

T[n_, k_] := Sum[Binomial[n, i] Binomial[n, k-2i], {i, 0, k/2}]; Table[T[n, k], {n, 0, 6}, {k, 0, 3n}] // Flatten (* Jean-François Alcover, Feb 02 2018 *)

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

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 24 06:13 EST 2018. Contains 299597 sequences. (Running on oeis4.)