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A008287
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Triangle of quadrinomial coefficients, row n is the sequence of coefficients of (1+x+x^2+x^3)^n
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21
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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; internal format)
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OFFSET
| 0,7
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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. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), 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]
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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).
Freund, J. E., Restricted Occupancy Theory - A Generalization of Pascal's Triangle, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 20-27.
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LINKS
| T. D. Noe, Rows n=0..25 of triangle, flattened
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)
W. Florek and T. Lulek, Combinatorial analysis of magnetic configurations
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
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FORMULA
| n-th row is formed by expanding (1+x+x^2+x^3)^n
Contribution of Vladimir Shevelev, Dec 15 2010: (Start)
q(n,0)=1; q(n,3*n)=1; q(n,k)=q(n,3*n-k);
q(n,k)=0, iff k<0 or k>3*n; Sum{k=0,...,n}q(n,k)=4^n; Sum{k=0,...,n}((-1)^k)*q(n,k)=0;
q(n,k)=Sum{i=0,...,floor(k/2)}Binom(n,i)*Binom(n,k-2*i);
q(n+1,k)=q(n,k-3)+q(n,k-2)+q(n,k-1)+q(n,k). (End)
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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; ...
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MATHEMATICA
| Flatten[Table[CoefficientList[(1 + x + x^2 + x^3)^n, x], {n, 0, 10}]] (* T. D. Noe, Apr 04 2011 *)
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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
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CROSSREFS
| Cf. A007318, A027907, A177882
Sequence in context: A017869 A107469 A167600 * A017859 A171456 A028356
Adjacent sequences: A008284 A008285 A008286 * A008288 A008289 A008290
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KEYWORD
| nonn,tabf,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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