login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A035343 Triangle of coefficients in expansion of (1 + x + x^2 + x^3 + x^4)^n. 23
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.

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 [0]  [1]  [2]  [3]  [4]  [5]  [6]  [7]  [8]  [9]  [10] [11] [12]

[0] 1;

[1] 1,   1,   1,   1,   1;

[2] 1,   2,   3,   4,   5,   4,   3,   2,   1;

[3] 1,   3,   6,   10,  15,  18,  19,  18,  15,  10,  6,   3,   1;

[4] ...

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

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 24 01:49 EST 2020. Contains 338603 sequences. (Running on oeis4.)