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A035343 Triangle of coefficients in expansion of (1 + x + x^2 + x^3 + x^4)^n.

%I A035343 #100 May 02 2025 10:30:10
%S A035343 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,
%T A035343 10,20,35,52,68,80,85,80,68,52,35,20,10,4,1,1,5,15,35,70,121,185,255,
%U A035343 320,365,381,365,320,255,185,121,70,35,15,5,1,1,6,21,56,126,246,426,666
%N A035343 Triangle of coefficients in expansion of (1 + x + x^2 + x^3 + x^4)^n.
%C A035343 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
%C A035343 The n-th row has 4n+1 terms (A016813). - _Michel Marcus_, Sep 08 2013
%C A035343 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
%C A035343 T(n,k) is the number of ways to obtain a sum of n+k when throwing a 5-sided die n times. - _Feryal Alayont_, Dec 30 2024
%D A035343 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77-78, 16. for q=5.
%D A035343 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).
%H A035343 T. D. Noe, <a href="/A035343/b035343.txt">Rows n = 0..25, flattened</a>
%H A035343 Moussa Ahmia and Hacene Belbachir, <a href="https://doi.org/10.37236/2255">Preserving log-convexity for generalized Pascal triangles</a>, Electronic Journal of Combinatorics, 19(2) (2012), #P16. - From _N. J. A. Sloane_, Oct 13 2012
%H A035343 Said Amrouche, Hacène Belbachir, <a href="https://arxiv.org/abs/2001.11665">Asymmetric extension of Pascal-Dellanoy triangles</a>, arXiv:2001.11665 [math.CO], 2020.
%H A035343 Armen G. Bagdasaryan, Ovidiu Bagdasar, <a href="https://doi.org/10.1016/j.endm.2018.05.012">On some results concerning generalized arithmetic triangles</a>, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.
%H A035343 Tomislav Došlić, <a href="https://doi.org/10.26493/1855-3974.1508.f8c">Block allocation of a sequential resource</a>, Ars Mathematica Contemporanea (2019) Vol. 17, 79-88.
%H A035343 Nour-Eddine Fahssi, <a href="http://arxiv.org/abs/1202.0228">Polynomial Triangles Revisited</a>, arXiv:1202.0228 [math.CO], (25-July-2012).
%H A035343 D. C. Fielder and C. O. Alford, <a href="/A027907/a027907_2.pdf">Pascal's triangle: top gun or just one of the gang?</a>, Applications of Fibonacci Numbers 4 (1991), 77-90. (Annotated scanned copy)
%H A035343 S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a>, arXiv:0802.2654 [math.NT], 2006.
%H A035343 J. E. Freund, <a href="http://www.jstor.org/stable/2308048">Restricted Occupancy Theory - A Generalization of Pascal's Triangle</a>, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 20-27.
%H A035343 Kuhapatanakul, Kantaphon; Anantakitpaisal, Pornpawee <a href="https://doi.org/10.1080/23311835.2017.1333293">The k-nacci triangle and applications</a>.  Cogent Math. 4, Article ID 1333293, 13 p. (2017).
%H A035343 Thorsten Neuschel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Neuschel/neuschel4.html">A Note on Extended Binomial Coefficients</a>, J. Int. Seq. 17 (2014) # 14.10.4.
%H A035343 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 4.
%H A035343 Eric Rowland, <a href="https://arxiv.org/abs/1704.05872">A matrix generalization of a theorem of Fine</a>, arXiv:1704.05872 [math.NT], 2017. See p.5.
%H A035343 Eric Rowland, <a href="http://math.colgate.edu/~integers/sjs18/sjs18.Abstract.html">A matrix generalization of a theorem of Fine</a>, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A18.
%H A035343 Bao-Xuan Zhu, <a href="http://arxiv.org/abs/1605.00257">Linear transformations and strong q-log-concavity for certain combinatorial triangle</a>, arXiv preprint arXiv:1605.00257 [math.CO], 2016.
%F A035343 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
%e A035343 Triangle begins:
%e A035343 n\k [0]  [1]  [2]  [3]  [4]  [5]  [6]  [7]  [8]  [9]  [10] [11] [12]
%e A035343 [0] 1;
%e A035343 [1] 1,   1,   1,   1,   1;
%e A035343 [2] 1,   2,   3,   4,   5,   4,   3,   2,   1;
%e A035343 [3] 1,   3,   6,   10,  15,  18,  19,  18,  15,  10,  6,   3,   1;
%e A035343 [4] ...
%p A035343 #Define the r-nomial coefficients for r = 1, 2, 3, ...
%p A035343 rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
%p A035343 #Display the 5-nomials as a table
%p A035343 r := 5:  rows := 10:
%p A035343 for n from 0 to rows do
%p A035343 seq(rnomial(r,n,k), k = 0..(r-1)*n)
%p A035343 end do;
%p A035343 # _Peter Bala_, Sep 07 2013
%t A035343 Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4)^n, x], {n, 0, 10}]] (* _T. D. Noe_, Apr 04 2011 *)
%o A035343 (Maxima) pentanomial(n,k):=coeff(expand((1+x+x^2+x^3+x^4)^n),x,k);
%o A035343 create_list(pentanomial(n,k),n,0,6,k,0,4*n); /* _Emanuele Munarini_, Mar 15 2011 */
%o A035343 (PARI) row(n) = Vec(((1 + x + x^2 + x^3 + x^4)^n) + O(x^(4*n+1)))
%o A035343 trianglerows(n) = for(k=0, n-1, print(row(k)))
%o A035343 /* Print initial 5 rows of triangle as follows */
%o A035343 trianglerows(5) \\ _Felix Fröhlich_, Aug 26 2018
%Y A035343 Cf. A007318, A027907, A008287. A063260, A063265, A171890, A213651, A213652.
%K A035343 nonn,tabf,easy
%O A035343 0,8
%A A035343 _N. J. A. Sloane_