A004320 -id:A004320 - cp's OEIS frontend

A000332 Binomial coefficient binomial(n,4) = n(n-1)(n-2)*(n-3)/24.

0, 0, 0, 0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, 40920, 46376, 52360, 58905, 66045, 73815, 82251, 91390, 101270, 111930, 123410

Offset: 0

N. J. A. Sloane

Number of intersection points of diagonals of convex n-gon where no more than two diagonals intersect at any point in the interior.
Also the number of equilateral triangles with vertices in an equilateral triangular array of points with n rows (offset 1), with any orientation. - Ignacio Larrosa Cañestro, Apr 09 2002. [See Les Reid link for proof. - N. J. A. Sloane, Apr 02 2016] [See Peter Kagey link for alternate proof. - Sameer Gauria, Jul 29 2025]
Start from cubane and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink on chemistry. - Robert G. Wilson v, Aug 02 2002
For n>0, a(n) = (-1/8)*(coefficient of x in Zagier's polynomial P_(2n,n)). (Zagier's polynomials are used by PARI/GP for acceleration of alternating or positive series.)
Figurate numbers based on the 4-dimensional regular convex polytope called the regular 4-simplex, pentachoron, 5-cell, pentatope or 4-hypertetrahedron with Schlaefli symbol {3,3,3}. a(n)=((n*(n-1)*(n-2)*(n-3))/4!). - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004, R. J. Mathar, Jul 07 2009
Maximal number of crossings that can be created by connecting n vertices with straight lines. - Cameron Redsell-Montgomerie (credsell(AT)uoguelph.ca), Jan 30 2007
If X is an n-set and Y a fixed (n-1)-subset of X then a(n) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Product of four consecutive numbers divided by 24. - Artur Jasinski, Dec 02 2007
The only prime in this sequence is 5. - Artur Jasinski, Dec 02 2007
For strings consisting entirely of 0's and 1's, the number of distinct arrangements of four 1's such that 1's are not adjacent. The shortest possible string is 7 characters, of which there is only one solution: 1010101, corresponding to a(5). An eight-character string has 5 solutions, nine has 15, ten has 35 and so on, congruent to A000332. - Gil Broussard, Mar 19 2008
For a(n)>0, a(n) is pentagonal if and only if 3 does not divide n. All terms belong to the generalized pentagonal sequence (A001318). Cf. A000326, A145919, A145920. - Matthew Vandermast, Oct 28 2008
Nonzero terms = row sums of triangle A158824. - Gary W. Adamson, Mar 28 2009
Except for the 4 initial 0's, is equivalent to the partial sums of the tetrahedral numbers A000292. - Jeremy Cahill (jcahill(AT)inbox.com), Apr 15 2009
If the first 3 zeros are disregarded, that is, if one looks at binomial(n+3, 4) with n>=0, then it becomes a 'Matryoshka doll' sequence with alpha=0: seq(add(add(add(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..50). - Peter Luschny, Jul 14 2009
For n>=1, a(n) is the number of n-digit numbers the binary expansion of which contains two runs of 0's. - Vladimir Shevelev, Jul 30 2010
For n>0, a(n) is the number of crossing set partitions of {1,2,..,n} into n-2 blocks. - Peter Luschny, Apr 29 2011
The Kn3, Ca3 and Gi3 triangle sums of A139600 are related to the sequence given above, e.g., Gi3(n) = 2*A000332(n+3) - A000332(n+2) + 7*A000332(n+1). For the definitions of these triangle sums, see A180662. - Johannes W. Meijer, Apr 29 2011
For n > 3, a(n) is the hyper-Wiener index of the path graph on n-2 vertices. - Emeric Deutsch, Feb 15 2012
Except for the four initial zeros, number of all possible tetrahedra of any size, having the same orientation as the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts. - V.J. Pohjola, Aug 31 2012
a(n+3) is the number of different ways to color the faces (or the vertices) of a regular tetrahedron with n colors if we count mirror images as the same.
a(n) = fallfac(n,4)/4! is also the number of independent components of an antisymmetric tensor of rank 4 and dimension n >= 1. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Does not satisfy Benford's law [Ross, 2012] - N. J. A. Sloane, Feb 12 2017
Number of chiral pairs of colorings of the vertices (or faces) of a regular tetrahedron with n available colors. Chiral colorings come in pairs, each the reflection of the other. - Robert A. Russell, Jan 22 2020
From Mircea Dan Rus, Aug 26 2020: (Start)
a(n+3) is the number of lattice rectangles (squares included) in a staircase of order n; this is obtained by stacking n rows of consecutive unit lattice squares, aligned either to the left or to the right, which consist of 1, 2, 3, ..., n squares and which are stacked either in the increasing or in the decreasing order of their lengths. Below, there is a staircase or order 4 which contains a(7) = 35 rectangles. [See the Teofil Bogdan and Mircea Dan Rus link, problem 3, under A004320]
    _
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(End)
a(n+4) is the number of strings of length n on an ordered alphabet of 5 letters where the characters in the word are in nondecreasing order. E.g., number of length-2 words is 15: aa,ab,ac,ad,ae,bb,bc,bd,be,cc,cd,ce,dd,de,ee. - Jim Nastos, Jan 18 2021
From Tom Copeland, Jun 07 2021: (Start)
Aside from the zeros, this is the fifth diagonal of the Pascal matrix A007318, the only nonvanishing diagonal (fifth) of the matrix representation IM = (A132440)^4/4! of the differential operator D^4/4!, when acting on the row vector of coefficients of an o.g.f., or power series.
M = e^{IM} is the matrix of coefficients of the Appell sequence p_n(x) = e^{D^4/4!} x^n = e^{b. D} x^n = (b. + x)^n = Sum_{k=0..n} binomial(n,k) b_n x^{n-k}, where the (b.)^n = b_n have the e.g.f. e^{b.t} = e^{t^4/4!}, which is that for A025036 aerated with triple zeros, the first column of M.
See A099174 and A000292 for analogous relationships for the third and fourth diagonals of the Pascal matrix. (End)
For integer m and positive integer r >= 3, the polynomial a(n) + a(n + m) + a(n + 2*m) + ... + a(n + r*m) in n has its zeros on the vertical line Re(n) = (3 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

			a(5) = 5 from the five independent components of an antisymmetric tensor A of rank 4 and dimension 5, namely A(1,2,3,4), A(1,2,3,5), A(1,2,4,5), A(1,3,4,5) and A(2,3,4,5). See the Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 70.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 294.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Charles W. Trigg, Mathematical Quickies, New York: Dover Publications, Inc., 1985, p. 53, #191.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 127.

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..1002
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Brandy Amanda Barnette, Counting Convex Sets on Products of Totally Ordered Sets, Masters Theses & Specialist Projects, Paper 1484, 2015.
Gaston A. Brouwer, Jonathan Joe, Abby A. Noble, and Matt Noble, Problems on the Triangular Lattice, arXiv:2405.12321 [math.CO], 2024. Mentions this sequence.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Paul Erdős, Norbert Kaufman, R. H. Koch, and Arthur Rosenthal, E750 (Interior diagonal points), Amer. Math. Monthly, 54 (Jun, 1947), p. 344.
Th. Grüner, A. Kerber, R. Laue, and M. Meringer, Mathematics for Combinatorial Chemistry.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 254.
Milan Janjic, Two Enumerative Functions.
Peter Kagey, A Proof Without Words: Triangles in the Triangular Grid, arXiv:2211.00186 [math.HO], 2022.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Iva Kodrnja and Helena Koncul, Polynomials vanishing on a basis of S_m(Gamma_0(N)), Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 324.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
Tim McDevitt and Kathryn Sutcliffe, A New Look at an Old Triangle Counting Problem. The Mathematics Teacher. Vol. 110, No. 6 (February 2017), pp. 470-474.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Les Reid, Counting Triangles in an Array.
Les Reid, Counting Triangles in an Array. [Cached copy]
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
Kirill S. Shardakov and Vladimir P. Bubnov, Stochastic Model of a High-Loaded Monitoring System of Data Transmission Network, Selected Papers of the Models and Methods of Information Systems Research Workshop, CEUR Workshop Proceedings, (St. Petersburg, Russia, 2019), 29-34.
Eric Weisstein's World of Mathematics, Composition.
Eric Weisstein's World of Mathematics, Pentatope Number.
Eric Weisstein's World of Mathematics, Pentatope.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index to sequences related to pyramidal numbers.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index entries for sequences related to Benford's law.

binomial(n, k): A161680 (k = 2), A000389 (k = 5), A000579 (k = 6), A000580 (k = 7), A000581 (k = 8), A000582 (k = 9).
Cf. A053134, A053126, A075733, A006322, A006325.
Cf. also A000583, A014820, A092181, A092182, A092183.
Cf. A000217, A000292, A007318 (column k = 4).
Cf. A158824.
Cf. A006008 (Number of ways to color the faces (or vertices) of a regular tetrahedron with n colors when mirror images are counted as two).
Cf. A104712 (third column, k=4).
See A269747 for a 3-D analog.
Cf. A006008 (oriented), A006003 (achiral) tetrahedron colorings.
Row 3 of A325000, col. 4 of A007318.
Cf. A000292, A025036, A099174.

A000332 := List([1..10^2], n -> Binomial(n, 4)); # Muniru A Asiru, Oct 16 2017

[Binomial(n,4): n in [0..50]]; // Vincenzo Librandi, Nov 23 2014

A000332 := n->binomial(n,4); [seq(binomial(n,4), n=0..100)];

Table[ Binomial[n, 4], {n, 0, 45} ] (* corrected by Harvey P. Dale, Aug 22 2011 *)
Table[(n-4)(n-3)(n-2)(n-1)/24, {n, 100}] (* Artur Jasinski, Dec 02 2007 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,0,0,0,1}, 45] (* Harvey P. Dale, Aug 22 2011 *)
CoefficientList[Series[x^4 / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 23 2014 *)

PARI
```
a(n)=binomial(n,4);
```

# Starts at a(3), i.e. computes n*(n+1)*(n+2)*(n+3)/24
# which is more in line with A000217 and A000292.
def A000332():
    x, y, z, u = 1, 1, 1, 1
    yield 0
    while True:
        yield x
        x, y, z, u = x + y + z + u + 1, y + z + u + 1, z + u + 1, u + 1
a = A000332(); print([next(a) for i in range(41)]) # Peter Luschny, Aug 03 2019

print([n*(n-1)*(n-2)*(n-3)//24 for n in range(50)])
# Gennady Eremin, Feb 06 2022

a(n) = n*(n-1)*(n-2)*(n-3)/24.
G.f.: x^4/(1-x)^5. - Simon Plouffe in his 1992 dissertation
a(n) = n*a(n-1)/(n-4). - Benoit Cloitre, Apr 26 2003, R. J. Mathar, Jul 07 2009
a(n) = Sum_{k=1..n-3} Sum_{i=1..k} i*(i+1)/2. - Benoit Cloitre, Jun 15 2003
Convolution of natural numbers {1, 2, 3, 4, ...} and A000217, the triangular numbers {1, 3, 6, 10, ...}. - Jon Perry, Jun 25 2003
a(n) = A110555(n+1,4). - Reinhard Zumkeller, Jul 27 2005
a(n+1) = ((n^5-(n-1)^5) - (n^3-(n-1)^3))/24 - (n^5-(n-1)^5-1)/30; a(n) = A006322(n-2)-A006325(n-1). - Xavier Acloque, Oct 20 2003; R. J. Mathar, Jul 07 2009
a(4*n+2) = Pyr(n+4, 4*n+2) where the polygonal pyramidal numbers are defined for integers A>2 and B>=0 by Pyr(A, B) = B-th A-gonal pyramid number = ((A-2)*B^3 + 3*B^2 - (A-5)*B)/6; For all positive integers i and the pentagonal number function P(x) = x*(3*x-1)/2: a(3*i-2) = P(P(i)) and a(3*i-1) = P(P(i) + i); 1 + 24*a(n) = (n^2 + 3*n + 1)^2. - Jonathan Vos Post, Nov 15 2004
First differences of A000389(n). - Alexander Adamchuk, Dec 19 2004
For n > 3, the sum of the first n-2 tetrahedral numbers (A000292). - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005 [Corrected by Doug Bell, Jun 25 2017]
Starting (1, 5, 15, 35, ...), = binomial transform of [1, 4, 6, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
Sum_{n>=4} 1/a(n) = 4/3, from the Taylor expansion of (1-x)^3*log(1-x) in the limit x->1. - R. J. Mathar, Jan 27 2009
A034263(n) = (n+1)*a(n+4) - Sum_{i=0..n+3} a(i). Also A132458(n) = a(n)^2 - a(n-1)^2 for n>0. - Bruno Berselli, Dec 29 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1. - Harvey P. Dale, Aug 22 2011
a(n) = (binomial(n-1,2)^2 - binomial(n-1,2))/6. - Gary Detlefs, Nov 20 2011
a(n) = Sum_{k=1..n-2} Sum_{i=1..k} i*(n-k-2). - Wesley Ivan Hurt, Sep 25 2013
a(n) = (A000217(A000217(n-2) - 1))/3 = ((((n-2)^2 + (n-2))/2)^2 - (((n-2)^2 + (n-2))/2))/(2*3). - Raphie Frank, Jan 16 2014
Sum_{n>=0} a(n)/n! = e/24. Sum_{n>=3} a(n)/(n-3)! = 73*e/24. See A067764 regarding the second ratio. - Richard R. Forberg, Dec 26 2013
Sum_{n>=4} (-1)^(n+1)/a(n) = 32*log(2) - 64/3 = A242023 = 0.847376444589... . - Richard R. Forberg, Aug 11 2014
4/(Sum_{n>=m} 1/a(n)) = A027480(m-3), for m>=4. - Richard R. Forberg, Aug 12 2014
E.g.f.: x^4*exp(x)/24. - Robert Israel, Nov 23 2014
a(n+3) = C(n,1) + 3*C(n,2) + 3*C(n,3) + C(n,4). Each term indicates the number of ways to use n colors to color a tetrahedron with exactly 1, 2, 3, or 4 colors.
a(n) = A080852(1,n-4). - R. J. Mathar, Jul 28 2016
From Gary W. Adamson, Feb 06 2017: (Start)
G.f.: Starting (1, 5, 14, ...), x/(1-x)^5 can be written
as (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = (1+x)^5;
as (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1+x+x^2)^5;
as (x * r(x) * r(x^4) * r(x^16) * r(x^64) * ...) where r(x) = (1+x+x^2+x^3)^5;
... (as a conjectured infinite set). (End)
From Robert A. Russell, Jan 22 2020: (Start)
a(n) = A006008(n) - a(n+3) = (A006008(n) - A006003(n)) / 2 = a(n+3) - A006003(n).
a(n+3) = A006008(n) - a(n) = (A006008(n) + A006003(n)) / 2 = a(n) + A006003(n).
a(n) = A007318(n,4).
a(n+3) = A325000(3,n). (End)
Product_{n>=5} (1 - 1/a(n)) = cosh(sqrt(15)*Pi/2)/(100*Pi). - Amiram Eldar, Jan 21 2021

Some formulas that referred to another offset corrected by R. J. Mathar, Jul 07 2009

A061356 Triangle read by rows: T(n, k) is the number of labeled trees on n nodes with maximal node degree k (0 < k < n).

Original entry on oeis.org

1, 2, 1, 9, 6, 1, 64, 48, 12, 1, 625, 500, 150, 20, 1, 7776, 6480, 2160, 360, 30, 1, 117649, 100842, 36015, 6860, 735, 42, 1, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1, 43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 1

Offset: 2

Olivier Gérard, Jun 07 2001

Essentially the coefficients of the Abel polynomials (A137452). - Peter Luschny, Jun 12 2022
This is a formula from Comtet, Theorem F, vol. I, p. 81 (French edition) used in proving Theorem D.
If we let N = n+1, binomial(N-2, k-1)*(N-1)^(N-k-1) = binomial(n-1, k-1)*n^(n-k), so this sequence with offset 1,1 also gives the number of rooted forests of k trees over [n]. - Washington Bomfim, Jan 09 2008
Let S(n,k) be the signed triangle, S(n,k) = (-1)^(n-k)T(n,k), which starts 1, -2, 1, 9, -6, 1, ..., then the inverse of S is the triangle of idempotent numbers A059298. - Peter Luschny, Mar 13 2009
With offset 1 also number of labeled multigraphs of k components, n nodes, and no cycles except one loop in each component. See link below to have a picture showing the bijection between rooted forests and multigraphs of this kind. (Note that there are no labels in the picture, but the bijection remains true if we label the nodes.) - Washington Bomfim, Sep 04 2010
With offset 1, T(n,k) is the number of forests of rooted trees on n nodes with exactly k (rooted) trees. - Geoffrey Critzer, Feb 10 2012
Also the Bell transform of the sequence (n+1)^n (A000169(n+1)) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016
Abel polynomials A(n,x) = x*(x+n)^(n-1) satisfy d/dx A(n,x) = n*A(n-1,x+1). - Michael Somos, May 10 2024
Also, T(n,k) is the number of parking functions with k ties. - Kyle Celano, Aug 18 2025

			Triangle begins
    1;
    2,     1;
    9,     6,     1;
   64,    48,    12,    1;
  625,   500,   150,   20,    1;
 7776,  6480,  2160,  360,   30,    1;
 ...
From _Peter Bala_, Sep 21 2012: (Start)
O.g.f.'s for the diagonals begin:
1/(1-x) = 1 + x + x^2 + x^3 + ...
2*x/(1-x)^3 = 2 + 6*x + 12*x^3 + ... A002378(n+1)
(9+3*x)/(1-x)^5 = 9 + 48*x + 150*x^2 + ... 3*A004320(n+1)
The numerator polynomials are the row polynomials of A155163.
(End)

L. Comtet, Analyse Combinatoire, P.U.F., Paris 1970. Volume 1, p 81.
L. Comtet, Advanced Combinatorics, Reidel, 1974.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
A. Avron and N. Dershowitz, Cayley's Formula, A Page From The Book, Amer. Math. Monthly, Vol. 123, No. 7, Aug.-Sep. 2016, 699-700 (2).
Peter Bala, Diagonals of triangles with generating function exp(t*F(x))
W. Bomfim, Bijection between rooted forests and multigraphs without cycles except one loop in each connected component. [From _Washington Bomfim_, Sep 04 2010]
J. W. Moon, Another proof of Cayley's formula for counting trees, Amer. Math. Monthly, Vol. 70, No. 8, Oct. 1963, 846-847.
Jim Pitman, Coalescent Random Forests, Technical Report No. 457, Department of Statistics, University of California.
Jim Pitman, Coalescent Random Forests, Journal of Combinatorial Theory, Series A, Volume 85, Issue 2, February 1999, Pages 165-193.
Paul R. F. Schumacher, Descents in parking functions, J. Integer Seq., Vol. 21 (2018), Article 18.2.3.
Eric Weisstein's World of Mathematics, Abel Polynomial.
Wikipedia, Lambert W function
J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 45-54.

Variant of A137452.
Columns are A000169, A053506, A053507, A053508, A053509.
First diagonal is A002378.
Row sums give A000272.
Cf. A028421, A059297, A139526 (row reverse), A155163, A202017.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0,...) as column 0 to the triangle.
BellMatrix(n -> (n+1)^n, 12); # Peter Luschny, Jan 21 2016

nn = 7; t = Sum[n^(n - 1)  x^n/n!, {n, 1, nn}]; f[list_] := Select[list, # > 0 &]; Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y t], {x, 0, nn}], {x, y}], 1]] // Flatten  (* Geoffrey Critzer, Feb 10 2012 *)
T[n_, m_] := T[n, m] = Binomial[n, m]*Sum[m^k*T[n-m, k], {k, 1, n-m}]; T[n_, n_] = 1; Table[T[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Vladimir Kruchinin *)
Table[Binomial[n - 2, k - 1]*(n - 1)^(n - k - 1), {n, 2, 12}, {k, 1, n - 1}] // Flatten (* G. C. Greubel, Nov 12 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(# + 1)^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

create_list(binomial(n,k)*(n+1)^(n-k),n,0,20,k,0,n); /* Emanuele Munarini, Apr 01 2014 */

for(n=2,11, for(k=1,n-1, print1(binomial(n-2, k-1)*(n-1)^(n-k-1), ", "))) \\ G. C. Greubel, Nov 12 2017

# uses[bell_matrix from A264428]
# Adds (1,0,0,0,...) as column 0 to the triangle.
bell_matrix(lambda n: (n+1)^n, 12) # Peter Luschny, Jan 21 2016

T(n, k) = binomial(n-2, k-1)*(n-1)^(n-k-1).
E.g.f.: (-LambertW(-y)/y)^(x+1)/(1+LambertW(-y)). - Vladeta Jovovic
From Peter Bala, Sep 21 2012: (Start)
Let T(x) = Sum_{n >= 0} n^(n-1)*x^n/n! denote the tree function of A000169. E.g.f.: F(x,t) := exp(t*T(x)) - 1 = -1 + {T(x)/x}^t = t*x + t*(2 + t)*x^2/2! + t*(9 + 6*t + t^2)*x^3/3! + ....
The compositional inverse with respect to x of (1/t)*F(x,t) is the e.g.f. for a signed version of the row reverse of A028421.
The row generating polynomials are the Abel polynomials A(n,x) = x*(x+n)^(n-1) for n >= 1.
Define B(n,x) = x^n/(1+n*x)^(n+1) = (-1)^n*A(-n,-1/x) for n >= 1. The k-th column entries are the coefficients in the formal series expansion of x^k in terms of B(n,x). For example, Col. 1: x = B(1,x) + 2*B(2,x) + 9*B(3,x) + 64*B(4,x) + ..., Col. 2: x^2 = B(2,x) + 6*B(3,x) + 48*B(4,x) + 500*B(5,x) + ... Compare with A059297.
n-th row sum = A000272(n+1).
Row reverse triangle is A139526.
The o.g.f.'s for the diagonals of the triangle are the rational functions R(n,x)/(1-x)^(2*n+1), where R(n,x) are the row polynomials of A155163. See below for examples.
(End)
T(n,m) = C(n,m)*Sum_{k=1..n-m} m^k*T(n-m,k), T(n,n) = 1. - Vladimir Kruchinin, Mar 31 2015

A233440 Triangle read by rows: T(n, k) = nbinomial(n, k)A000757(k), 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 0, 0, 3, 4, 0, 0, 16, 4, 5, 0, 0, 50, 25, 40, 6, 0, 0, 120, 90, 288, 216, 7, 0, 0, 245, 245, 1176, 1764, 1603, 8, 0, 0, 448, 560, 3584, 8064, 14656, 13000, 9, 0, 0, 756, 1134, 9072, 27216, 74196, 131625, 118872, 10, 0, 0, 1200, 2100, 20160, 75600, 274800, 731250, 1320800, 1202880

Offset: 0

Luis Manuel Rivera Martínez, Dec 09 2013

For n >= 0, 0 <= k <= n, T(n, k) is the number of permutations of n symbols that k-commute with an n-cycle (we say that two permutations f and g k-commute if H(fg, gf) = k, where H(, ) denotes the Hamming distance between permutations).

Row sums give A000142.

			For n = 4 and k = 4, T(4, 4) = 4 because all the permutations of 4 symbols that 4-commute with permutation (1, 2, 3, 4) are (1, 3), (2, 4), (1, 2)(3, 4) and (1, 4)(2, 3).

Luis Manuel Rivera Martínez, Rows n = 0..30 of triangle, flattened
R. Moreno and L. M. Rivera, Blocks in cycles and k-commuting permutations, arXiv:1306.5708 [math.CO], 2013-2014.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A007318, A000757.

T[n_, k_] := n Binomial[n, k] ((-1)^k+Sum[(-1)^j k!/(k-j)/j!, {j, 0, k-1}]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = n*C(n,k)*A000757(k), 0 <= k <= n.
Bivariate e.g.f.: G(z, u) = z*exp(z*(1-u))*(u/(1-z*u)+(1-log(1-z*u))*(1-u)).
T(n, 0)  = A001477(n), n>=0;
T(n, 1)  = A000004(n), n>=1;
T(n, 2)  = A000004(n), n>=2;
T(n, 3)  = A004320(n-2), n>=3;
T(n, 4)  = A027764(n-1), n>=4;
T(n, 5)  = A027765(n-1)*A000757(5), n>=5;
T(n, 6)  = A027766(n-1)*A000757(6), n>=6;
T(n, 7)  = A027767(n-1)*A000757(7), n>=7;
T(n, 8)  = A027768(n-1)*A000757(8), n>=8;
T(n, 9)  = A027769(n-1)*A000757(9), n>=9;
T(n, 10) = A027770(n-1)*A000757(10), n>=10;
T(n, 11) = A027771(n-1)*A000757(11), n>=11;
T(n, 12) = A027772(n-1)*A000757(12), n>=12;
T(n, 13) = A027773(n-1)*A000757(13), n>=13;
T(n, 14) = A027774(n-1)*A000757(14), n>=14;
T(n, 15) = A027775(n-1)*A000757(15), n>=15;
T(n, 16) = A027776(n-1)*A000757(16), n>=16. - Luis Manuel Rivera Martínez, Feb 08 2014
T(n, 0)+T(n, 3) = n*A050407(n+1), for n>=0. - Luis Manuel Rivera Martínez, Mar 06 2014

A206817 Sum_{0

Original entry on oeis.org

1, 10, 73, 520, 3967, 33334, 309661, 3166468, 35416555, 430546642, 5655609529, 79856902816, 1206424711303, 19419937594990, 331860183278677, 6000534640290364, 114462875817046051, 2297294297649673738, 48394006967070653425

Offset: 2

Clark Kimberling, Feb 12 2012

nonn

In the following guide to related sequences,

c(n) = Sum_{0

t(n) = Sum_{0

s(k).................c(n)........t(n)

k....................A000217.....A000292

k^2..................A016061.....A004320

k^3..................A206808.....A206809

k^4..................A206810.....A206811

k!...................A206816.....A206817

prime(k).............A152535.....A062020

prime(k+1)...........A185382.....A206803

2^(k-1)..............A000337.....A045618

k(k+1)/2.............A007290.....A034827

k-th quarter-square..A049774.....A206806

			a(3) = (2-1) + (6-1) + (6-2) = 10.

Danny Rorabaugh, Table of n, a(n) for n = 2..400

Cf. A000142, A206816.

s[k_] := k!; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1];
Table[c[n], {n, 2, 32}]          (* A206816 *)
Flatten[Table[t[n], {n, 2, 20}]] (* A206817 *)

a(n)=sum(j=1,n,j!*(2*j-n-1)) \\ Charles R Greathouse IV, Oct 11 2015

a(n)=my(t=1); sum(j=1,n,t*=j; t*(2*j-n-1)) \\ Charles R Greathouse IV, Oct 11 2015

[sum([sum([factorial(k)-factorial(j) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,21)] # Danny Rorabaugh, Apr 18 2015

a(n) = a(n-1)+(n-1)s(n)-p(n-1), where s(n) = n! and p(k) = 1!+2!+...+k!.

a(n) = Sum_{k=2..n} A206816(k).

A047929 a(n) = n^2(n-1)(n-2).

Original entry on oeis.org

0, 18, 96, 300, 720, 1470, 2688, 4536, 7200, 10890, 15840, 22308, 30576, 40950, 53760, 69360, 88128, 110466, 136800, 167580, 203280, 244398, 291456, 345000, 405600, 473850, 550368, 635796, 730800, 836070, 952320, 1080288, 1220736

Offset: 2

N. J. A. Sloane

There are 5 ways to put parentheses in the expression a - b - c - d: (a - (b - c)) - d, ((a - b) - c) - d, (a - b) - (c - d), a - (b - (c - d)), a - ((b - c) - d). This sequence describes how many sets of natural numbers [a,b,c,d] can be produced with the numbers {0,1,2,3,...,n} such that all the distinct expressions take different values. A045991 describes the similar process for a - b - c. For example, no sets can be produced with only 0's or only 0's and 1's; with {0,1,2,3}, 18 such sets can be produced. - Asher Auel, Jan 26 2000

For n >= 3, a(n)/6 is the number of permutations of n symbols that 3-commute with an n-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for sequences related to parenthesizing
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A004320, A045991, A233440.

[n^2*(n-1)*(n-2): n in [2..40]]; // Vincenzo Librandi, May 02 2011

Drop[CoefficientList[Series[6 x^3*(3 + x)/(1 - x)^5, {x, 0, 34}], x], 2] (* Michael De Vlieger, May 21 2021 *)

a(n)=n^4 - 3*n^3 + 2*n^2 \\ Charles R Greathouse IV, May 02 2011

a(n) = A004320(n-2)*6.
G.f.: 6*x^3*(3 + x)/(1 - x)^5. - Stefano Spezia, May 20 2021
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, May 22 2021
From Amiram Eldar, May 25 2021: (Start)
Sum_{n>=3} 1/a(n) = (Pi^2 - 9)/12.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 + 2*log(2) - 7/4. (End)

A060556 Bisection of triangle A060098: odd-indexed members of column sequences of A060098 (not counting leading zeros).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 12, 16, 4, 1, 20, 50, 32, 5, 1, 30, 120, 140, 55, 6, 1, 42, 245, 448, 316, 86, 7, 1, 56, 448, 1176, 1284, 622, 126, 8, 1, 72, 756, 2688, 4170, 3102, 1113, 176, 9, 1, 90, 1200, 5544, 11550, 12122

Offset: 0

Wolfdieter Lang, Apr 06 2001

Row sums give A060557. Column sequences without leading zeros give for m=0..5: A000012 (powers of 1), A002378 = 2*A000217, A004320, 4*A040977, A060558, 2*A060559.
Companion triangle (even-indexed members) A060102.
With offset 1 for n and k, T(n,k) is the number of (1-2-3)-avoiding trapezoidal words of length n that contain n+1-k 1s. A trapezoidal word (following Riordan) is a sequence (a_1,a_2,...,a_n) of integers with 1 <= a_i <= 2i-1. For example, T(3,3)=3 counts 122, 132, 133 and T(4,2)=12 counts 1112, 1113, 1114, 1115, 1116, 1117, 1121, 1131, 1141, 1151, 1211, 1311. - David Callan, Aug 25 2009

			{1}; {1,2}; {1,6,3}; {1,12,16,4}; ...; Po(3,x) = 3 + x.

a(n, m)= A060098(2*n+1-m, m).
G.f. for column m: (x^m)*Po(m+1, x)/(1-x)^(2*m+1), with Po(n, x) = Sum_{j=0..floor(n/2)} binomial(n, 2*j+1)*x^j (odd members of row n of Pascal triangle A007318).
a(n, m) = Sum_{j=0..floor((m+1)/2)} binomial(n-j+m, 2*m)*binomial(m+1, 2*j+1), n >= m >= 0, otherwise zero.

A024197 a(n) = 3rd elementary symmetric function of the first n+2 odd positive integers.

Original entry on oeis.org

15, 176, 950, 3480, 10045, 24640, 53676, 106800, 197835, 345840, 576290, 922376, 1426425, 2141440, 3132760, 4479840, 6278151, 8641200, 11702670, 15618680, 20570165, 26765376, 34442500, 43872400, 55361475, 69254640, 85938426, 105844200, 129451505

Offset: 1

Clark Kimberling

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Contribution from Johannes W. Meijer, Jun 08 2009: (Start)
Equals fourth right hand column of A028338 triangle.
Equals fourth left hand column of A109692 triangle.
Equals fourth right hand column of A161198 triangle divided by 2^m.
(End)

Vec(-x*(x^3+33*x^2+71*x+15)/(x-1)^7 + O(x^100)) \\ Colin Barker, Aug 15 2014

a(n) = n*(n+1)*(n+2)^2*(n^2+3*n+1)/6.
G.f.: -x*(x^3+33*x^2+71*x+15) / (x-1)^7. - Colin Barker, Aug 15 2014
a(n) = A004320(n)*A028387(n). - R. J. Mathar, Oct 01 2016
a(n) = A028338(n+2, n-1), n >= 1, (fourth diagonal). See a crossref. below. - Wolfdieter Lang, Jul 21 2017

A330805 Number of squares and rectangles in the interior of the square with vertices (n,0), (0,n), (-n,0) and (0,-n) in a square (x,y)-grid.

Original entry on oeis.org

0, 9, 51, 166, 410, 855, 1589, 2716, 4356, 6645, 9735, 13794, 19006, 25571, 33705, 43640, 55624, 69921, 86811, 106590, 129570, 156079, 186461, 221076, 260300, 304525, 354159, 409626, 471366, 539835, 615505, 698864, 790416, 890681, 1000195, 1119510, 1249194, 1389831

Offset: 0

Luce ETIENNE, Jan 01 2020

Collection: 2*n*(n+1)-ominoes.
Number of squares (all sizes): (8*n^3 + 24*n^2 + 22*n - 3*(-1)^n + 3)/12.
Number of rectangles (all sizes): (8*n^4 + 24*n^3 + 22*n^2 + 3*(-1)^n - 3)/12.

			a(1) = 4*1+5 = 9; a(2) = 4*5+31 = 51; a(3) = 4*15 + 106 = 166; a(4) = 4*36 + 270 = 410.

Teofil Bogdan and Mircea Dan Rus, Counting the lattice rectangles inside Aztec diamonds and square biscuits, arXiv:2007.13472 [math.CO], 2020.
Luce ETIENNE, Illustration of a(1), a(2) and a(3).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000332, A004320, A046092, A111746, A212523.

LinearRecurrence[{5,-10,10,-5,1},{0,9,51,166,410},40] (* Harvey P. Dale, Jun 27 2020 *)

G.f.: x*(x + 3)^2/(1 - x)^5.
E.g.f.: (1/6)*exp(x)*x*(54 + 99*x + 40*x^2 + 4*x^3). - Stefano Spezia, Jan 01 2020
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = n*(n + 1)*(4*n^2 + 12*n + 11)/6.
a(n) = 4*A000332(n+3) + A212523(n+1).
a(n) = 9*A000332(n+3) + 6*A000332(n+2) + A000332(n+1). - Mircea Dan Rus, Aug 26 2020
a(n) = 3*A004320(n) + A004320(n-1). - Mircea Dan Rus, Aug 26 2020

A060558 Fifth column (m=4) of triangle A060556.

Original entry on oeis.org

5, 55, 316, 1284, 4170, 11550, 28380, 63492, 131703, 256685, 474760, 839800, 1429428, 2352732, 3759720, 5852760, 8900265, 13252899, 19362596, 27804700, 39303550, 54761850, 75294180, 102265020

Offset: 0

Wolfdieter Lang, Apr 06 2001

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A000217, A004320, A040977.

a(n)={(2*n^2 + 18*n + 35)*binomial(n + 6, 6)/7} \\ Harry J. Smith, Jul 07 2009

a(n) = (2*n^2 + 18*n + 35)*binomial(n+6, 6)/7.

G.f.: (5 + 10*x + x^2)/(1-x)^9.

A185733 Second accumulation array of the polygonal number array (A086270), by antidiagonals.

Original entry on oeis.org

1, 5, 3, 15, 16, 6, 35, 50, 34, 10, 70, 120, 110, 60, 15, 126, 245, 270, 200, 95, 21, 210, 448, 560, 500, 325, 140, 28, 330, 756, 1036, 1050, 825, 490, 196, 36, 495, 1200, 1764, 1960, 1750, 1260, 700, 264, 45, 715, 1815, 2820, 3360, 3290, 2695, 1820, 960, 345, 55, 1001, 2640, 4290, 5400, 5670, 5096, 3920, 2520, 1275, 440, 66, 1365, 3718, 6270, 8250, 9150, 8820, 7448, 5460, 3375, 1650, 550, 78

Offset: 1

Clark Kimberling, Feb 02 2011

This is the accumulation array of the accumulation array of A086270. The accumulation array of A185733 is A185734, so that A184733 is the weight array of A185734. Thus, the arrays are members of a two-way infinite chain; see A144112 for definitions.

			Northwest corner:
1....5....15....35....70
3....16...50....120...245
6....34...110...270...560
10...60...200...500..1050

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows 1 to 2: A000332, A004320.
Columns 1 to 3: A000219, A096941, 5*A007603.
Cf. A144112, A086270, A185732, A184734.

f[n_,k_]:=k*(1+k)*(2+k)*n*(1+n)*(10+2*k-n+k*n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k*(k+1)*(k+2)*n*(n+1)*(k*n-n+2*k+10)/144.

Showing 1-10 of 15 results. Next

cp's OEIS Frontend

A000332 Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.

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A061356 Triangle read by rows: T(n, k) is the number of labeled trees on n nodes with maximal node degree k (0 < k < n).

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A233440 Triangle read by rows: T(n, k) = n*binomial(n, k)*A000757(k), 0 <= k <= n.

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A206817 Sum_{0

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A047929 a(n) = n^2*(n-1)*(n-2).

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A060556 Bisection of triangle A060098: odd-indexed members of column sequences of A060098 (not counting leading zeros).

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A000332 Binomial coefficient binomial(n,4) = n(n-1)(n-2)*(n-3)/24.

A233440 Triangle read by rows: T(n, k) = nbinomial(n, k)A000757(k), 0 <= k <= n.

A047929 a(n) = n^2(n-1)(n-2).