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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 Alois P. Heinz, Rows n = 0..100, flattened (first 26 rows from T. D. Noe) 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 Armen G. Bagdasaryan, Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77. 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. 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 T. Neuschel, A Note on Extended Binomial Coefficients, J. Int. Seq. 17 (2014) # 14.10.4. 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 # second Maple program: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))((1+x+x^2+x^3)^n): seq(T(n), n=0..10);  # Alois P. Heinz, Aug 17 2018 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] *)  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 STATUS approved

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Last modified June 25 09:48 EDT 2019. Contains 324347 sequences. (Running on oeis4.)