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 A035343 Triangle of coefficients in expansion of (1 + x + x^2 + x^3 + x^4)^n. 27
 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 246, 426, 666 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Coefficient of x^k in (1 + x + x^2 + x^3 + x^4)^n is the number of distinct ways in which k unlabeled objects can be distributed in n labeled urns allowing at most 4 objects to fall in each urn. - N-E. Fahssi, Mar 16 2008 The n-th row has 4n+1 terms (A016813). - Michel Marcus, Sep 08 2013 Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (1,2), (1,3), (1,4). - Nicholas Ham, Sep 14 2018 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77-78, 16. for q=5. 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, flattened Moussa Ahmia and Hacene Belbachir, Preserving log-convexity for generalized Pascal triangles, Electronic Journal of Combinatorics, 19(2) (2012), #P16. - From N. J. A. Sloane, Oct 13 2012 Said Amrouche, Hacène Belbachir, Asymmetric extension of Pascal-Dellanoy triangles, arXiv:2001.11665 [math.CO], 2020. Armen G. Bagdasaryan, Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77. Tomislav Došlić, Block allocation of a sequential resource, Ars Mathematica Contemporanea (2019) Vol. 17, 79-88. 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) S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2006. J. E. Freund, Restricted Occupancy Theory - A Generalization of Pascal's Triangle, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 20-27. Kuhapatanakul, Kantaphon; Anantakitpaisal, Pornpawee The k-nacci triangle and applications. Cogent Math. 4, Article ID 1333293, 13 p. (2017). T. Neuschel, A Note on Extended Binomial Coefficients, J. Int. Seq. 17 (2014) # 14.10.4. Eric Rowland, A matrix generalization of a theorem of Fine, arXiv:1704.05872 [math.NT], 2017. See p.5. Eric Rowland, A matrix generalization of a theorem of Fine, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A18. Bao-Xuan Zhu, Linear transformations and strong q-log-concavity for certain combinatorial triangle, arXiv preprint arXiv:1605.00257 [math.CO], 2016. FORMULA T(n,k) = Sum_{i = 0..floor(k/5)} (-1)^i*binomial(n,i)*binomial(n+k-1-5*i,n-1) for n >= 0 and 0 <= k <= 4*n. - Peter Bala, Sep 07 2013 EXAMPLE Triangle begins: n\k               1;  1, 1, 1, 1, 1;  1, 2, 3, 4, 5, 4, 3, 2, 1;  1, 3, 6, 10, 15, 18, 19, 18, 15, 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 5-nomials as a table r := 5: 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 + x^4)^n, x], {n, 0, 10}]] (* T. D. Noe, Apr 04 2011 *) PROG (Maxima) pentanomial(n, k):=coeff(expand((1+x+x^2+x^3+x^4)^n), x, k); create_list(pentanomial(n, k), n, 0, 6, k, 0, 4*n); \\ Emanuele Munarini, Mar 15 2011 (PARI) row(n) = Vec(((1 + x + x^2 + x^3 + x^4)^n) + O(x^(4*n+1))) trianglerows(n) = for(k=0, n-1, print(row(k))) /* Print initial 5 rows of triangle as follows */ trianglerows(5) \\ Felix Fröhlich, Aug 26 2018 CROSSREFS Cf. A007318, A027907, A008287. A063260, A063265, A171890, A213651, A213652. Sequence in context: A017890 A134011 A280913 * A017880 A086144 A131974 Adjacent sequences: A035340 A035341 A035342 * A035344 A035345 A035346 KEYWORD nonn,tabf,easy AUTHOR STATUS approved

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Last modified March 31 14:41 EDT 2023. Contains 361661 sequences. (Running on oeis4.)