A050534 -id:A050534 - cp's OEIS frontend

A000292 Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n(n+1)(n+2)/6.

0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180

Offset: 0

N. J. A. Sloane

a(n) is the number of balls in a triangular pyramid in which each edge contains n balls.
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012).
Also (1/6)*(n^3 + 3*n^2 + 2*n) is the number of ways to color the vertices of a triangle using <= n colors, allowing rotations and reflections. Group is the dihedral group D_6 with cycle index (x1^3 + 2*x3 + 3*x1*x2)/6.
Also the convolution of the natural numbers with themselves. - Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001
Connected with the Eulerian numbers (1, 4, 1) via 1*a(n-2) + 4*a(n-1) + 1*a(n) = n^3. - Gottfried Helms, Apr 15 2002
a(n) is sum of all the possible products p*q where (p,q) are ordered pairs and p + q = n + 1. E.g., a(5) = 5 + 8 + 9 + 8 + 5 = 35. - Amarnath Murthy, May 29 2003
Number of labeled graphs on n+3 nodes that are triangles. - Jon Perry, Jun 14 2003
Number of permutations of n+3 which have exactly 1 descent and avoid the pattern 1324. - Mike Zabrocki, Nov 05 2004
Schlaefli symbol for this polyhedron: {3,3}.
Transform of n^2 under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005
a(n) is a perfect square only for n = {1, 2, 48}. E.g., a(48) = 19600 = 140^2. - Alexander Adamchuk, Nov 24 2006
a(n+1) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4)^n. - Sergio Falcon, Feb 12 2007 [Corrected by Graeme McRae, Aug 28 2007]
a(n+1) is the number of terms in the complete homogeneous symmetric polynomial of degree n in 3 variables. - Richard Barnes, Sep 06 2017
This is also the average "permutation entropy", sum((pi(n)-n)^2)/n!, over the set of all possible n! permutations pi. - Jeff Boscole (jazzerciser(AT)hotmail.com), Mar 20 2007
a(n) = (d/dx)(S(n, x), x)|A049310.%20-%20_Wolfdieter%20Lang">{x = 2}. First derivative of Chebyshev S-polynomials evaluated at x = 2. See A049310. - _Wolfdieter Lang, Apr 04 2007
If X is an n-set and Y a fixed (n-1)-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Complement of A145397; A023533(a(n))=1; A014306(a(n))=0. - Reinhard Zumkeller, Oct 14 2008
Equals row sums of triangle A152205. - Gary W. Adamson, Nov 29 2008
a(n) is the number of gifts received from the lyricist's true love up to and including day n in the song "The Twelve Days of Christmas". a(12) = 364, almost the number of days in the year. - Bernard Hill (bernard(AT)braeburn.co.uk), Dec 05 2008
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF2 denominators of A156925. See A157703 for background information. - Johannes W. Meijer, Mar 07 2009
Starting with 1 = row sums of triangle A158823. - Gary W. Adamson, Mar 28 2009
Wiener index of the path with n edges. - Eric W. Weisstein, Apr 30 2009
This is a 'Matryoshka doll' sequence with alpha=0, the multiplicative counterpart is A000178: seq(add(add(i,i=alpha..k),k=alpha..n),n=alpha..50). - Peter Luschny, Jul 14 2009
a(n) is the number of nondecreasing triples of numbers from a set of size n, and it is the number of strictly increasing triples of numbers from a set of size n+2. - Samuel Savitz, Sep 12 2009 [Corrected and enhanced by Markus Sigg, Sep 24 2023]
a(n) is the number of ordered sequences of 4 nonnegative integers that sum to n. E.g., a(2) = 10 because 2 = 2 + 0 + 0 + 0 = 1 + 1 + 0 + 0 = 0 + 2 + 0 + 0 = 1 + 0 + 1 + 0 = 0 + 1 + 1 + 0 = 0 + 0 + 2 + 0 = 1 + 0 + 0 + 1 = 0 + 1 + 0 + 1 = 0 + 0 + 1 + 1 = 0 + 0 + 0 + 2. - Artur Jasinski, Nov 30 2009
a(n) corresponds to the total number of steps to memorize n verses by the technique described in A173964. - Ibrahima Faye (ifaye2001(AT)yahoo.fr), Feb 22 2010
The number of (n+2)-bit numbers which contain two runs of 1's in their binary expansion. - Vladimir Shevelev, Jul 30 2010
a(n) is also, starting at the second term, the number of triangles formed in n-gons by intersecting diagonals with three diagonal endpoints (see the first column of the table in Sommars link). - Alexandre Wajnberg, Aug 21 2010
Column sums of:
1 4 9 16 25...
    1  4  9...
          1...
..............
--------------
1 4 10 20 35...
From Johannes W. Meijer, May 20 2011: (Start)
The Ca3, Ca4, Gi3 and Gi4 triangle sums (see A180662 for their definitions) of the Connell-Pol triangle A159797 are linear sums of shifted versions of the duplicated tetrahedral numbers, e.g., Gi3(n) = 17*a(n) + 19*a(n-1) and Gi4(n) = 5*a(n) + a(n-1).
Furthermore the Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above. (End)
a(n-2)=N_0(n), n >= 1, with a(-1):=0, is the number of vertices of n planes in generic position in three-dimensional space. See a comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p. 506. - Wolfdieter Lang, May 27 2011
We consider optimal proper vertex colorings of a graph G. Assume that the labeling, i.e., coloring starts with 1. By optimality we mean that the maximum label used is the minimum of the maximum integer label used across all possible labelings of G. Let S=Sum of the differences |l(v) - l(u)|, the sum being over all edges uv of G and l(w) is the label associated with a vertex w of G. We say G admits unique labeling if all possible labelings of G is S-invariant and yields the same integer partition of S. With an offset this sequence gives the S-values for the complete graph on n vertices, n = 2, 3, ... . - K.V.Iyer, Jul 08 2011
Central term of commutator of transverse Virasoro operators in 4-D case for relativistic quantum open strings (ref. Zwiebach). - Tom Copeland, Sep 13 2011
Appears as a coefficient of a Sturm-Liouville operator in the Ovsienko reference on page 43. - Tom Copeland, Sep 13 2011
For n > 0: a(n) is the number of triples (u,v,w) with 1 <= u <= v <= w <= n, cf. A200737. - Reinhard Zumkeller, Nov 21 2011
Regarding the second comment above by Amarnath Murthy (May 29 2003), see A181118 which gives the sequence of ordered pairs. - L. Edson Jeffery, Dec 17 2011
The dimension of the space spanned by the 3-form v[ijk] that couples to M2-brane worldsheets wrapping 3-cycles inside tori (ref. Green, Miller, Vanhove eq. 3.9). - Stephen Crowley, Jan 05 2012
a(n+1) is the number of 2 X 2 matrices with all terms in {0, 1, ..., n} and (sum of terms) = n. Also, a(n+1) is the number of 2 X 2 matrices with all terms in {0, 1, ..., n} and (sum of terms) = 3*n. - Clark Kimberling, Mar 19 2012
Using n + 4 consecutive triangular numbers t(1), t(2), ..., t(n+4), where n is the n-th term of this sequence, create a polygon by connecting points (t(1), t(2)) to (t(2), t(3)), (t(2), t(3)) to (t(3), t(4)), ..., (t(1), t(2)) to (t(n+3), t(n+4)). The area of this polygon will be one-half of each term in this sequence. - J. M. Bergot, May 05 2012
Pisano period lengths:  1, 4, 9, 8, 5, 36, 7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17,108, 19, 40, ... . (The Pisano sequence modulo m is the auxiliary sequence p(n) = a(n) mod m, n >= 1, for some m. p(n) is periodic for all sequences with rational g.f., like this one, and others. The lengths of the period of p(n) are quoted here for m>=1.) - R. J. Mathar, Aug 10 2012
a(n) is the maximum possible number of rooted triples consistent with any phylogenetic tree (level-0 phylogenetic network) containing exactly n+2 leaves. - Jesper Jansson, Sep 10 2012
For n > 0, the digital roots of this sequence A010888(a(n)) form the purely periodic 27-cycle {1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 9, 9, 9}, which just rephrases the Pisano period length above. - Ant King, Oct 18 2012
a(n) is the number of functions f from {1, 2, 3} to {1, 2, ..., n + 4} such that f(1) + 1 < f(2) and f(2) + 1 < f(3). - Dennis P. Walsh, Nov 27 2012
a(n) is the Szeged index of the path graph with n+1 vertices; see the Diudea et al. reference, p. 155, Eq. (5.8). - Emeric Deutsch, Aug 01 2013
Also the number of permutations of length n that can be sorted by a single block transposition. - Vincent Vatter, Aug 21 2013
From J. M. Bergot, Sep 10 2013: (Start)
a(n) is the 3 X 3 matrix determinant
  |  C(n,1)   C(n,2)   C(n,3)  |
  | C(n+1,1) C(n+1,2) C(n+1,3) |
  | C(n+2,1) C(n+2,2) C(n+2,3) |
(End)
In physics, a(n)/2 is the trace of the spin operator S_z^2 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and the sum of their squares is 10/2 = a(3)/2. - Stanislav Sykora, Nov 06 2013
a(n+1) = (n+1)*(n+2)*(n+3)/6 is also the dimension of the Hilbert space of homogeneous polynomials of degree n. - L. Edson Jeffery, Dec 12 2013
For n >= 4, a(n-3) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0111 (n-4 zeros), or, equivalently, a(n-3) is up-down coefficient {n,7} (see comment in A060351). - Vladimir Shevelev, Feb 15 2014
a(n) is one-half the area of the region created by plotting the points (n^2,(n+1)^2). A line connects points (n^2,(n+1)^2) and ((n+1)^2, (n+2)^2) and a line is drawn from (0,1) to each increasing point. From (0,1) to (4,9) the area is 2; from (0,1) to (9,16) the area is 8; further areas are 20,40,70,...,2*a(n). - J. M. Bergot, May 29 2014
Beukers and Top prove that no tetrahedral number > 1 equals a square pyramidal number A000330. - Jonathan Sondow, Jun 21 2014
a(n+1) is for n >= 1 the number of nondecreasing n-letter words over the alphabet [4] = {1, 2, 3, 4} (or any other four distinct numbers). a(2+1) = 10 from the words 11, 22, 33, 44, 12, 13, 14, 23, 24, 34; which is also the maximal number of distinct elements in a symmetric 4 X 4 matrix. Inspired by the Jul 20 2014 comment by R. J. Cano on A000582. - Wolfdieter Lang, Jul 29 2014
Degree of the q-polynomial counting the orbits of plane partitions under the action of the symmetric group S3. Orbit-counting generating function is Product_{i <= j <= k <= n} ( (1 - q^(i + j + k - 1))/(1 - q^(i + j + k - 2)) ). See q-TSPP reference. - Olivier Gérard, Feb 25 2015
Row lengths of tables A248141 and A248147. - Reinhard Zumkeller, Oct 02 2014
If n is even then a(n) = Sum_{k=1..n/2} (2k)^2. If n is odd then a(n) = Sum_{k=0..(n-1)/2} (1+2k)^2. This can be illustrated as stacking boxes inside a square pyramid on plateaus of edge lengths 2k or 2k+1, respectively. The largest k are the 2k X 2k or (2k+1) X (2k+1) base. - R. K. Guy, Feb 26 2015
Draw n lines in general position in the plane. Any three define a triangle, so in all we see C(n,3) = a(n-2) triangles (6 lines produce 4 triangles, and so on). - Terry Stickels, Jul 21 2015
a(n-2) = fallfac(n,3)/3!, n >= 3, is also the number of independent components of an antisymmetric tensor of rank 3 and dimension n. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015
Number of compositions (ordered partitions) of n+3 into exactly 4 parts. - Juergen Will, Jan 02 2016
Number of weak compositions (ordered weak partitions) of n-1 into exactly 4 parts. - Juergen Will, Jan 02 2016
For n >= 2 gives the number of multiplications of two nonzero matrix elements in calculating the product of two upper n X n triangular matrices. - John M. Coffey, Jun 23 2016
Terms a(4n+1), n >= 0, are odd, all others are even. The 2-adic valuation of the subsequence of every other term, a(2n+1), n >= 0, yields the ruler sequence A007814. Sequence A275019 gives the 2-adic valuation of a(n). - M. F. Hasler, Dec 05 2016
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 12 2017
C(n+2,3) is the number of ways to select 1 triple among n+2 objects, thus a(n) is the coefficient of x1^(n-1)*x3 in exponential Bell polynomial B_{n+2}(x1,x2,...), hence its link with A050534 and A001296 (see formula). - Cyril Damamme, Feb 26 2018
a(n) is also the number of 3-cycles in the (n+4)-path complement graph. - Eric W. Weisstein, Apr 11 2018
a(n) is the general number of all geodetic graphs of diameter n homeomorphic to a complete graph K4. - Carlos Enrique Frasser, May 24 2018
a(n) + 4*a(n-1) + a(n-2) = n^3 = A000578(n), for n >= 0 (extending the a(n) formula given in the name). This is the Worpitzky identity for cubes. (Number of components of the decomposition of a rank 3 tensor in dimension n >= 1 into symmetric, mixed and antisymmetric parts). For a(n-2) see my Dec 10 2015 comment. - Wolfdieter Lang, Jul 16 2019
a(n) also gives the total number of regular triangles of length k (in some length unit), with k from {1, 2, ..., n}, in the matchstick arrangement with enclosing triangle of length n, but only triangles with the orientation of the enclosing triangle are counted. Row sums of unsigned A122432(n-1, k-1), for n >= 1. See the Andrew Howroyd comment in A085691. - Wolfdieter Lang, Apr 06 2020
a(n) is the number of bigrassmannian permutations on n+1 elements, i.e., permutations which have a unique left descent, and a unique right descent. - Rafael Mrden, Aug 21 2020
a(n-2) is the number of chiral pairs of colorings of the edges or vertices of a triangle using n or fewer colors. - Robert A. Russell, Oct 20 2020
a(n-2) is the number of subsets of {1,2,...,n} whose diameters are their size. For example, for n=4, a(2)=4 and the sets are {1,3}, {2,4}, {1,2,4}, {1,3,4}. - Enrique Navarrete, Dec 26 2020
For n>1, a(n-2) is the number of subsets of {1,2,...,n} in which the second largest element is the size of the subset. For example, for n=4, a(2)=4 and the sets are {2,3}, {2,4}, {1,3,4}, {2,3,4}. - Enrique Navarrete, Jan 02 2021
a(n) is the number of binary strings of length n+2 with exactly three 0's. - Enrique Navarrete, Jan 15 2021
From Tom Copeland, Jun 07 2021: (Start)
Aside from the zero, this sequence is the fourth diagonal of the Pascal matrix A007318 and the only nonvanishing diagonal (fourth) of the matrix representation IM = (A132440)^3/3! of the differential operator D^3/3!, when acting on the row vector of coefficients of an o.g.f., or power series.
M = e^{IM} is the lower triangular matrix of coefficients of the Appell polynomial sequence p_n(x) = e^{D^3/3!} 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^3/3!}, which is that for A025035 aerated with double zeros, the first column of M.
See A099174 and A000332 for analogous relationships for the third and fifth diagonals of the Pascal matrix. (End)
a(n) is the number of circles with a radius of integer length >= 1 and center at a grid point in an n X n grid. - Albert Swafford, Jun 11 2021
Maximum Wiener index over all connected graphs with n+1 vertices. - Allan Bickle, Jul 09 2022
The third Euler row (1,4,1) has an additional connection with the tetrahedral numbers besides the n^3 identity stated above: a^2(n) + 4*a^2(n+1) + a^2(n+2) = a(n^2+4n+4), which can be shown with algebra. E.g., a^2(2) + 4*a^2(3) + a^2(4) = 16 + 400 + 400 = a(16). Although an analogous thing happens with the (1,1) row of Euler's triangle and triangular numbers C(n+1,2) = A000217(n) = T(n), namely both T(n-1) + T(n) = n^2 and T^2(n-1) + T^2(n) = T(n^2) are true, only one (the usual identity) still holds for the Euler row (1,11,11,1) and the C(n,4) numbers in A000332. That is, the dot product of (1,11,11,1) with the squares of 4 consecutive terms of A000332 is not generally a term of A000332. - Richard Peterson, Aug 21 2022
For n > 1, a(n-2) is the number of solutions of the Diophantine equation x1 + x2 + x3 + x4 + x5 = n, subject to the constraints 0 <= x1, 1 <= x2, 2 <= x3, 0 <= x4 <= 1, 0 <= x5 and x5 is even. - Daniel Checa, Nov 03 2022
a(n+1) is also the number of vertices of the generalized Pitman-Stanley polytope with parameters 2, n, and vector (1,1, ... ,1), which is integrally equivalent to a flow polytope over the grid graph having 2 rows and n columns. - William T. Dugan, Sep 18 2023
a(n) is the number of binary words of length (n+1) containing exactly one substring 01. a(2) = 4: 001, 010, 011, 101. - Nordine Fahssi, Dec 09 2024
a(n) is the number of directed bishop moves on an n X n chessboard, identified under rotations (0, 90, 180 and 270 degree) and all reflections. - Hilko Koning, Aug 27 2025

			a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of balls, 6 in a triangle at the bottom, 3 in the middle layer and 1 on top.
Consider the square array
  1  2  3  4  5  6 ...
  2  4  6  8 10 12 ...
  3  6  9 12 16 20 ...
  4  8 12 16 20 24 ...
  5 10 15 20 25 30 ...
  ...
then a(n) = sum of n-th antidiagonal. - _Amarnath Murthy_, Apr 06 2003
G.f. = x + 4*x^2 + 10*x^3 + 20*x^4 + 35*x^5 + 56*x^6 + 84*x^7 + 120*x^8 + 165*x^9 + ...
Example for a(3+1) = 20 nondecreasing 3-letter words over {1,2,3,4}: 111, 222, 333; 444, 112, 113, 114, 223, 224, 122, 224, 133, 233, 144, 244, 344; 123, 124, 134, 234.  4 + 4*3 + 4 = 20. - _Wolfdieter Lang_, Jul 29 2014
Example for a(4-2) = 4 independent components of a rank 3 antisymmetric tensor A of dimension 4: A(1,2,3), A(1,2,4), A(1,3,4) and A(2,3,4). - _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.
V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_0.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 44, 70.
H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
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. 4.
M. V. Diudea, I. Gutman, and J. Lorentz, Molecular Topology, Nova Science, 2001, Huntington, N.Y. pp. 152-156.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, pp. 292-293.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
V. Ovsienko and S. Tabachnikov, Projective Differential Geometry Old and New, Cambridge Tracts in Mathematics (no. 165), Cambridge Univ. Press, 2005.
Kenneth A Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42. doi:10.4169/math.mag.85.1.36.
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).
A. Szenes, The combinatorics of the Verlinde formulas (N.J. Hitchin et al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 11-13.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 126-127.
B. Zwiebach, A First Course in String Theory, Cambridge, 2004; see p. 226.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
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].
O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 5.
Nicolay Avilov, Process of emergence of a(5)
F. Beukers and J. Top, On oranges and integral points on certain plane cubic curves, Nieuw Arch. Wiskd., IV (1988), Ser. 6, No. 3, 203-210.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
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.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
William Dowling and Nadia Lafreniere, Homomesy on permutations with toggling actions, arXiv:2312.02383 [math.CO], 2023. See page 8.
W. T. Dugan, M. Hegarty, A. H. Morales, and A. Raymond, Generalized Pitman-Stanley polytope: vertices and faces, arXiv:2307.09925 [math.CO], 2023.
Gennady Eremin, Naturalized bracket row and Motzkin triangle, arXiv:2004.09866 [math.CO], 2020.
C. E. Frasser and G. N. Vostrov, Geodetic Graphs Homeomorphic to a Given Geodetic Graph, arXiv:1611.01873 [cs.DM], 2016. [p. 16, corollary 5]
Michael B. Green, Stephen D. Miller, and Pierre Vanhove, Small representations, string instantons, and Fourier modes of Eisenstein series, arXiv:1111.2983 [hep-th], 2011-2013.
N. Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
N. Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
Jacob Hicks, M. A. Ollis, and John. R. Schmitt, Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches, arXiv:1809.02684 [math.CO], 2018.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 46. Book's website
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
C. Homberger and V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv preprint arXiv:1308.4946 [math.CO], 2013.
Milan Janjic, Two Enumerative Functions
Virginia Johnson and Charles K. Cook, Areas of Triangles and other Polygons with Vertices from Various Sequences, arXiv:1608.02420 [math.CO], 2016.
R. Jovanovic, First 2500 Tetrahedral numbers
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
M. Kobayashi, Enumeration of bigrassmannian permutations below a permutation in Bruhat order, arXiv:1005.3335 [math.CO], 2011; Order 28(1) (2011), 131-137.
C. Koutschan, M. Kauers, and D. Zeilberger, A Proof Of George Andrews' and David Robbins' q-TSPP Conjecture, Proc. Nat. Acad. Sc., vol. 108 no. 6 (2011), pp. 2196-2199. See also Zeilberger's comments on this article; Local copy of comments (pdf file).
T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
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.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See page 6.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (1).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Valentin Ovsienko, Shadow sequences of integers, from Fibonacci to Markov and back, arXiv:2111.02553 [math.CO], 2021.
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.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Claude-Alexandre Simonetti, A new mathematical symbol : the termirial, arXiv:2005.00348 [math.GM], 2020.
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, Pyramid of 20 balls corresponding to a(3)=20.
S. E. Sommars and T. Sommars, Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, J. Integer Sequences, 1 (1998), #98.1.5.
H. Stamm-Wilbrandt, Sum of Pascal's triangle reciprocals
G. Villemin's Almanach of Numbers, Nombres Tétraédriques (in French).
Eric Weisstein's World of Mathematics, Composition
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Path Complement Graph
Eric Weisstein's World of Mathematics, Path Graph
Eric Weisstein's World of Mathematics, Tetrahedral Number
Eric Weisstein's World of Mathematics, Wiener Index
Yue Zhang, Chunfang Zheng, and David Sankoff, Distinguishing successive ancient polyploidy levels based on genome-internal syntenic alignment, BMC Bioinformatics (2019) Vol. 20, 635.
A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.
Index entries for "core" sequences
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for two-way infinite sequences
Index entries for sequences related to Benford's law

Bisections give A000447 and A002492.
Sums of 2 consecutive terms give A000330.
a(3n-3) = A006566(n). A000447(n) = a(2n-2). A002492(n) = a(2n+1).
Column 0 of triangle A094415.
Cf. A000217 (first differences), A001044,  (see above example), A061552, A040977, A133111, A133112, A152205, A158823, A156925, A157703, A173964, A058187, A190717, A190718, A100440, A181118, A222716.
Partial sums are A000332. - Jonathan Vos Post, Mar 27 2011
Cf. A216499 (the analogous sequence for level-1 phylogenetic networks).
Cf. A068980 (partitions), A231303 (spin physics).
Cf. similar sequences listed in A237616.
Cf. A104712 (second column, if offset is 2).
Cf. A145397 (non-tetrahedral numbers). - Daniel Forgues, Apr 11 2015
Cf. A127324.
Cf. A007814, A275019 (2-adic valuation).
Cf. A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
Cf. A002817 (4-cycle count of \bar P_{n+4}), A060446 (5-cycle count of \bar P_{n+3}), A302695 (6-cycle count of \bar P_{n+5})
Row 2 of A325000 (simplex facets and vertices) and A327084 (simplex edges and ridges).
Cf. A085691 (matchsticks), A122432 (unsigned row sums).
Cf. (triangle colorings) A006527 (oriented), A000290 (achiral), A327085 (chiral simplex edges and ridges).
Row 3 of A321791 (cycles of n colors using k or fewer colors).
Cf. A007318, A025035, A099174.
The Wiener indices of powers of paths for k = 1..6 are given in A000292, A002623, A014125, A122046, A122047, and A175724, respectively.

a:=n->Binomial(n+2,3);; A000292:=List([0..50],n->a(n)); # Muniru A Asiru, Feb 28 2018

a000292 n = n * (n + 1) * (n + 2) `div` 6
a000292_list = scanl1 (+) a000217_list
-- Reinhard Zumkeller, Jun 16 2013, Feb 09 2012, Nov 21 2011

[n*(n+1)*(n+2)/6: n in [0..50]]; // Wesley Ivan Hurt, Jun 03 2014

a:=n->n*(n+1)*(n+2)/6; seq(a(n), n=0..50);
A000292 := n->binomial(n+2,3); seq(A000292(n), n=0..50);
isA000292 := proc(n)
    option remember;
    local a,i ;
    for i from iroot(6*n,3)-1 do
        a := A000292(i) ;
        if a > n then
            return false;
        elif a = n then
            return true;
        end if;
    end do:
end proc: # R. J. Mathar, Aug 14 2024

Table[Binomial[n + 2, 3], {n, 0, 20}] (* Zerinvary Lajos, Jan 31 2010 *)
Accumulate[Accumulate[Range[0, 50]]] (* Harvey P. Dale, Dec 10 2011 *)
Table[n (n + 1)(n + 2)/6, {n,0,100}] (* Wesley Ivan Hurt, Sep 25 2013 *)
Nest[Accumulate, Range[0, 50], 2] (* Harvey P. Dale, May 24 2017 *)
Binomial[Range[20] + 1, 3] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 4, 10}, 20] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[x/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)
Table[Range[n].Range[n,1,-1],{n,0,50}] (* Harvey P. Dale, Mar 02 2024 *)

A000292(n):=n*(n+1)*(n+2)/6$ makelist(A000292(n),n,0,60); /* Martin Ettl, Oct 24 2012 */

a(n) = (n) * (n+1) * (n+2) / 6  \\ corrected by Harry J. Smith, Dec 22 2008

a=vector(10000);a[2]=1;for(i=3,#a,a[i]=a[i-2]+i*i); \\ Stanislav Sykora, Nov 07 2013

is(n)=my(k=sqrtnint(6*n,3)); k*(k+1)*(k+2)==6*n \\ Charles R Greathouse IV, Dec 13 2016

# Compare A000217.
def A000292():
    x, y, z = 1, 1, 1
    yield 0
    while True:
        yield x
        x, y, z = x + y + z + 1, y + z + 1, z + 1
a = A000292(); print([next(a) for i in range(45)]) # Peter Luschny, Aug 03 2019

a(n) = C(n+2,3) = n*(n+1)*(n+2)/6 (see the name).
G.f.: x / (1 - x)^4.
a(n) = -a(-4 - n) for all in Z.
a(n) = Sum_{k=0..n} A000217(k) = Sum_{k=1..n} Sum_{j=0..k} j, partial sums of the triangular numbers.
a(2n)= A002492(n). a(2n+1)=A000447(n+1).
a(n) = Sum_{1 <= i <= j <= n} |i - j|. - Amarnath Murthy, Aug 05 2002
a(n) = (n+3)*a(n-1)/n. - Ralf Stephan, Apr 26 2003
Sums of three consecutive terms give A006003. - Ralf Stephan, Apr 26 2003
Determinant of the n X n symmetric Pascal matrix M_(i, j) = C(i+j+2, i). - Benoit Cloitre, Aug 19 2003
The sum of a series constructed by the products of the index and the length of the series (n) minus the index (i): a(n) = sum[i(n-i)]. - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005
a(n) = Sum_{k=0..floor((n-1)/2)} (n-2k)^2 [offset 0]; a(n+1) = Sum_{k=0..n} k^2*(1-(-1)^(n+k-1))/2 [offset 0]. - Paul Barry, Apr 16 2005
a(n) = -A108299(n+5, 6) = A108299(n+6, 7). - Reinhard Zumkeller, Jun 01 2005
a(n) = -A110555(n+4, 3). - Reinhard Zumkeller, Jul 27 2005
Values of the Verlinde formula for SL_2, with g = 2: a(n) = Sum_{j=1..n-1} n/(2*sin^2(j*Pi/n)). - Simone Severini, Sep 25 2006
a(n-1) = (1/(1!*2!))*Sum_{1 <= x_1, x_2 <= n} |det V(x_1, x_2)| = (1/2)*Sum_{1 <= i,j <= n} |i-j|, where V(x_1, x_2) is the Vandermonde matrix of order 2. Column 2 of A133112. - Peter Bala, Sep 13 2007
Starting with 1 = binomial transform of [1, 3, 3, 1, ...]; e.g., a(4) = 20 = (1, 3, 3, 1) dot (1, 3, 3, 1) = (1 + 9 + 9 + 1). - Gary W. Adamson, Nov 04 2007
a(n) = A006503(n) - A002378(n). - Reinhard Zumkeller, Sep 24 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Jaume Oliver Lafont, Nov 18 2008
Sum_{n>=1} 1/a(n) = 3/2, case x = 1 in Gradstein-Ryshik 1.513.7. - R. J. Mathar, Jan 27 2009
E.g.f.:((x^3)/6 + x^2 + x)*exp(x). - Geoffrey Critzer, Feb 21 2009
Limit_{n -> oo} A171973(n)/a(n) = sqrt(2)/2. - Reinhard Zumkeller, Jan 20 2010
With offset 1, a(n) = (1/6)*floor(n^5/(n^2 + 1)). - Gary Detlefs, Feb 14 2010
a(n) = Sum_{k = 1..n} k*(n-k+1). - Vladimir Shevelev, Jul 30 2010
a(n) = (3*n^2 + 6*n + 2)/(6*(h(n+2) - h(n-1))), n > 0, where h(n) is the n-th harmonic number. - Gary Detlefs, Jul 01 2011
a(n) = coefficient of x^2 in the Maclaurin expansion of 1 + 1/(x+1) + 1/(x+1)^2 + 1/(x+1)^3 + ... + 1/(x+1)^n. - Francesco Daddi, Aug 02 2011
a(n) = coefficient of x^4 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - Francesco Daddi, Aug 04 2011
a(n) = 2*A002415(n+1)/(n+1). - Tom Copeland, Sep 13 2011
a(n) = A004006(n) - n - 1. - Reinhard Zumkeller, Mar 31 2012
a(n) = (A007531(n) + A027480(n) + A007290(n))/11. - J. M. Bergot, May 28 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 1. - Ant King, Oct 18 2012
G.f.: x*U(0) where U(k) = 1 + 2*x*(k+2)/( 2*k+1 - x*(2*k+1)*(2*k+5)/(x*(2*k+5)+(2*k+2)/U(k+1) )); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n^2 - 1) = (1/2)*(a(n^2 - n - 2) + a(n^2 + n - 2)) and
a(n^2 + n - 2) - a(n^2 - 1) = a(n-1)*(3*n^2 - 2) = 10*A024166(n-1), by Berselli's formula in A222716. - Jonathan Sondow, Mar 04 2013
G.f.: x + 4*x^2/(Q(0)-4*x) where Q(k) = 1 + k*(x+1) + 4*x - x*(k+1)*(k+5)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n+1) = det(C(i+3,j+2), 1 <= i,j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013
a(n) = a(n-2) + n^2, for n > 1. - Ivan N. Ianakiev, Apr 16 2013
a(2n) = 4*(a(n-1) + a(n)), for n > 0. - Ivan N. Ianakiev, Apr 26 2013
G.f.: x*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + (k+1)/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) = n + 2*a(n-1) - a(n-2), with a(0) = a(-1) = 0. - Richard R. Forberg, Jul 11 2013
a(n)*(m+1)^3 + a(m)*(n+1) = a(n*m + n + m), for any nonnegative integers m and n. This is a 3D analog of Euler's theorem about triangular numbers, namely t(n)*(2m+1)^2 + t(m) = t(2nm + n + m), where t(n) is the n-th triangular number. - Ivan N. Ianakiev, Aug 20 2013
Sum_{n>=0} a(n)/(n+1)! = 2*e/3 = 1.8121878856393... .  Sum_{n>=1} a(n)/n! = 13*e/6 = 5.88961062832... .  - Richard R. Forberg, Dec 25 2013
a(n+1) = A023855(n+1) + A023856(n). - Wesley Ivan Hurt, Sep 24 2013
a(n) = A024916(n) + A076664(n), n >= 1. - Omar E. Pol, Feb 11 2014
a(n) = A212560(n) - A059722(n). - J. M. Bergot, Mar 08 2014
Sum_{n>=1} (-1)^(n + 1)/a(n) = 12*log(2) - 15/2 = 0.8177661667... See A242024, A242023. - Richard R. Forberg, Aug 11 2014
3/(Sum_{n>=m} 1/a(n)) = A002378(m), for m > 0. - Richard R. Forberg, Aug 12 2014
a(n) = Sum_{i=1..n} Sum_{j=i..n} min(i,j). - Enrique Pérez Herrero, Dec 03 2014
Arithmetic mean of Square pyramidal number and Triangular number: a(n) = (A000330(n) + A000217(n))/2. - Luciano Ancora, Mar 14 2015
a(k*n) = a(k)*a(n) + 4*a(k-1)*a(n-1) + a(k-2)*a(n-2). - Robert Israel, Apr 20 2015
Dirichlet g.f.: (zeta(s-3) + 3*zeta(s-2) + 2*zeta(s-1))/6. - Ilya Gutkovskiy, Jul 01 2016
a(n) = A080851(1,n-1) - R. J. Mathar, Jul 28 2016
a(n) = (A000578(n+1) - (n+1) ) / 6. - Zhandos Mambetaliyev, Nov 24 2016
G.f.: x/(1 - x)^4 = (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...), where r(x) = (1 + x)^4 = (1 + 4x + 6x^2 + 4x^3 + x^4); and x/(1 - x)^4 = (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1 + x + x^2)^4. - Gary W. Adamson, Jan 23 2017
a(n) = A000332(n+3) - A000332(n+2). - Bruce J. Nicholson, Apr 08 2017
a(n) = A001296(n) - A050534(n+1). - Cyril Damamme, Feb 26 2018
a(n)  = Sum_{k=1..n} (-1)^(n-k)*A122432(n-1, k-1), for n >= 1, and a(0) = 0. - Wolfdieter Lang, Apr 06 2020
From Robert A. Russell, Oct 20 2020: (Start)
a(n) = A006527(n) - a(n-2) = (A006527(n) + A000290(n)) / 2 = a(n-2) + A000290(n).
a(n-2) = A006527(n) - a(n) = (A006527(n) - A000290(n)) / 2 = a(n) - A000290(n).
a(n) = 1*C(n,1) + 2*C(n,2) + 1*C(n,3), where the coefficient of C(n,k) is the number of unoriented triangle colorings using exactly k colors.
a(n-2) = 1*C(n,3), where the coefficient of C(n,k) is the number of chiral pairs of triangle colorings using exactly k colors.
a(n-2) = A327085(2,n). (End)
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(sqrt(2)*Pi)/(3*sqrt(2)*Pi).
Product_{n>=2} (1 - 1/a(n)) = sqrt(2)*sinh(sqrt(2)*Pi)/(33*Pi). (End)
a(n) = A002623(n-1) + A002623(n-2), for n>1. - Ivan N. Ianakiev, Nov 14 2021

Corrected and edited by Daniel Forgues, May 14 2010

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.

Original entry on oeis.org

0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413

Offset: 0

Clark Kimberling, Apr 27 2012

nonn

Each sequence in the following guide counts 4-tuples
(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}.  The notation "m div" means that m divides every term of the sequence.
A211058 ... wx <= yz
A211787 ... wx <= 2yz
A211795 ... wx < 2yz
A211797 ... wx > 2yz
A211809 ... wx >= 2yz
A211812 ... wx <= 3yz
A211917 ... wx < 3yz
A211918 ... wx > 3yz
A211919 ... wx >= 3yz
A211920 ... 2wx < 3yz
A211921 ... 2wx <= 3yz
A211922 ... 2wx > 3yz
A211923 ... 2wx >= 3yz
A212019 ... wx = 2yz ..... 2 div
A212020 ... wx = 3yz ..... 2 div
A212021 ... 2wx = 3yz .... 2 div
A212047 ... wx = 4yz
A212048 ... 3wx = 4yz .... 2 div
A212049 ... wx = 5yz ..... 2 div
A212050 ... 2wx = 5yz .... 2 div
A212051 ... 3wx = 5yz .... 2 div
A212052 ... 4wx = 5yz .... 2 div
A209978 ... wx = yz + 1 .. 2 div
A212053 ... wx <= yz + 1
A212054 ... wx > yz + 1
A212055 ... wx <= yz + 2
A212056 ... wx > yz + 2
A197168 ... wx = yz + 2 .. 2 div
A061201 ... w = xyz
A212057 ... w < xyz
A212058 ... w >= xyz
A212059 ... w = xyz - 1
A212060 ... w = xyz - 2
A212061 ... wx = (yz)^2
A212062 ... w^2 = xyz
A212063 ... w^2 < xyz
A212064 ... w^2 >= xyz
A212065 ... w^2 <= xyz
A212066 ... w^2 > xyz
A212067 ... w^3 = xyz
A002623 ... w = 2x + y + z
A006918 ... w = 2x + 2y + z
A000601 ... w = x + 2y + 3z (except for initial 0's)
A212068 ... 2w = x + y + z
A212069 ... 3w = x + y + z (w = average{x,y,z})
A212088 ... 3w < x + y + z
A212089 ... 3w >= x + y + z
A212090 ... w < x + y + z
A000332 ... w >= x + y + z
A212145 ... w < 2x + y + z
A001752 ... w >= 2x + y + z
A001400 ... w = 2x +3y + 4z
A005900 ... w = -x + y + z
A192023 ... w = -x + y + z + 2
A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div
A212087 ... w^2 + x^2 = y^2 + z^2
A212092 ... w^2 < x^2 + y^2 + z^2
A212093 ... w^2 <= x^2 + y^2 + z^2
A212094 ... w^2 > x^2 + y^2 + z^2
A212095 ... w^2 >= x^2 + y^2 + z^2
A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div
A212097 ... w^3 < x^3 + y^3 + z^3
A212098 ... w^3 <= x^3 + y^3 + z^3
A212099 ... w^3 > x^3 + y^3 + z^3
A212100 ... w^3 >= x^3 + y^3 + z^3
A212101 ... wx^2 = yz^2
A212102 ... 1/w = 1/x + 1/y + 1/z
A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}
A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.
A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.
A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.
A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.
A212133 ... median(w,x,y,z) = mean(w,x,y,z)
A212134 ... median(w,x,y,z) <= mean(w,x,y,z)
A212135 ... median(w,x,y,z) > mean(w,x,y,z)
A212241 ... wx + yz > n
A212243 ... 2wx + yz = n
A212244 ... w = xyz - n
A212245 ... w = xyz - 2n
A212246 ... 2w = x + y + z - n
A212247 ... 3w = x + y + z + n
A212249 ... 3w < x + y + z + n
A212250 ... 3w >= x + y + z + n
A212251 ... 3w = x + y + z + n + 1
A212252 ... 3w = x + y + z + n + 2
A212254 ... w = x + 2y + 3z - n
A212255 ... w^2 = mean(x^2, y^2, z^2)
A212256 ... 4/w = 1/x + 1/y +1/z + 1/n
In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent.  In the list below, the same is true for expressions involving more than one relation.
A000332 ... w < x <= y < z .... C(n,4)
A000914 ... w < x <= y < z .... Stirling 1st kind
A000914 ... w < x <= y >= z ... Stirling 1st kind
A050534 ... w < x < y >= z .... tritriangular
A001296 ... w <= x <= y >= z .. 4-dim pyramidal
A006322 ... x < x > y >= z
A002418 ... w < x >= y < z
A050534 ... w < x >=y >= z
A212415 ... w < x >= y <= z
A001296 ... w < x >= y <= z
A212246 ... w <= x > y <= z
A006322 ... w <= x >= y <= z
A212501 ... w > x < y >= z
A212503 ... w < 2x and y < 2z ..... A (note below)
A212504 ... w < 2x and y > 2z ..... A
A212505 ... w < 2x and y >= 2z .... A
A212506 ... w <= 2x and y <= 2z ... A
A212507 ... w < 2x and y <= 2z .... B
A212508 ... w < 2x and y < 3z ..... C
A212509 ... w < 2x and y <= 3z .... C
A212510 ... w < 2x and y > 3z ..... C
A212511 ... w < 2x and y >= 3z .... C
A212512 ... w <= 2x and y < 3z .... C
A212513 ... w <= 2x and y <= 3z ... C
A212514 ... w <= 2x and y > 3z .... C
A212515 ... w <= 2x and y >= 3z ... C
A212516 ... w > 2x and y < 3z ..... C
A212517 ... w > 2x and y <= 3z .... C
A212518 ... w > 2x and y > 3z ..... C
A212519 ... w > 2x and y >= 3z .... C
A212520 ... w >= 2x and y < 3z .... C
A212521 ... w >= 2x and y <= 3z ... C
A212522 ... w >= 2x and y > 3z .... C
A212523 ... w + x < y + z
A212560 ... w + x <= y + z
A212561 ... w + x = 2y + 2z
A212562 ... w + x < 2y + 2z ....... B
A212563 ... w + x <= 2y + 2z ...... B
A212564 ... w + x > 2y + 2z ....... B
A212565 ... w + x >= 2y + 2z ...... B
A212566 ... w + x = 3y + 3z
A212567 ... 2w + 2x = 3y + 3z
A212570 ... |w - x| = |x - y| + |y - z|
A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div
A212572 ... |w - x| <= |x - y| + |y - z| .. B
A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div
A212574 ... |w - x| >= |x - y| + |y - z| .. B
A212575 ... 2|w - x| = |x - y| + |y - z|
A212576 ... |w - x| = 2|x - y| + 2|y - z|
A212577 ... |w - x| = 2|x - y| - |y - z|
A212578 ... 2|w - x| = |x - y| - |y - z|
A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}
A212692 ... w = |x - y| + |y - z| ............... 2 div
A212568 ... w < |x - y| + |y - z| ............... 2 div
A212573 ... w <= |x - y| + |y - z| .............. 2 div
A212574 ... w > |x - y| + |y - z|
A212575 ... w >= |x - y| + |y - z|
A212676 ... w + x = |x - y| + |y - z| ......... H
A212677 ... w + y = |x - y| + |y - z|
A212678 ... w + x + y = |x - y| + |y - z|
A006918 ... w + x + y + z = |x - y| + |y - z| . H
A212679 ... |x - y| = |y - z| ................. H
A212680 ... |x - y| = |y - z| + 1 ..............H 2 div
A212681 ... |x - y| < |y - z| ................... 2 div
A212682 ... |x - y| >= |y - z|
A212683 ... |x - y| = w + |y - z| ............... 2 div
A212684 ... |x - y| = n - w + |y - z|
A212685 ... |w - x| = w + |y - z|
A186707 ... |w - x| < w + |y - z| ... (Note D)
A212714 ... |w - x| >= w + |y - z| .......... H . 2 div
A212686 ... 2*|w - x| = n + |y - z| ............. 4 div
A212687 ... 2*|w - x| < n + |y - z| ......... B
A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div
A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div
A212690 ... 2*|w - x| <= n + |y - z| ........ B
A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div
...
In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.
  R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.
A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A
A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A
A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A
A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div
A212744 ... w=range (=max-min) ............... E
A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}
A212746 ... R is in {w,x,y,z} ................ E
A212569 ... R is not in {w,x,y,z} ............ E
A212749 ... w=R or xA212750 ... w=R or x=R or yA212751 ... w=R or x=R or yA212752 ... wR ......... A
A212753 ... wR or z>R ......... D
A212754 ... wR or y>R or z>R ......... D
A002415 ... w = x + R ........................ D
A212755 ... |w - x| = R ...................... D
A212756 ... 2w = x + R
A212757 ... 2w = R
A212758 ... w = floor(R/2)
A002413 ... w = floor((x+y+z/2))
A212759 ... w, x, y are even
A212760 ... w is even and x = y + z .......... E
A212761 ... w is odd and x and y are even .... F . 2 div
A212762 ... w and x are odd y is even ........ F . 2 div
A212763 ... w, x, y are odd .................. F
A212764 ... w, x, y are even and z is odd .... F
A030179 ... w and x are even and y and z odd
A212765 ... w is even and x,y,z are odd ...... F
A212766 ... w is even and x is odd ........... A . 2 div
A212767 ... w and x are even and w+x=y+z ..... E
A212889 ... R is even ........................ A
A212890 ... R is odd ......................... A . 2 div
A212742 ... w-x, x-y, y-z are all even ....... A
A212892 ... w-x, x-y, y-z are all odd ........ A
A212893 ... w-x, x-y, y-z have same parity ... A
A005915 ... min{|w-x|, |x-y|, |y-z|} = 0
A212894 ... min{|w-x|, |x-y|, |y-z|} = 1
A212895 ... min{|w-x|, |x-y|, |y-z|} = 2
A179824 ... min{|w-x|, |x-y|, |y-z|} > 0
A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1
A212897 ... min{|w-x|, |x-y|, |y-z|} > 1
A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2
A212899 ... min{|w-x|, |x-y|, |y-z|} > 2
A212901 ... |w-x| = |x-y| = |y-z|
A212900 ... |w-x|, |x-y|, |y-z| are distinct . G
A212902 ... |w-x| < |x-y| < |y-z| ............ G
A212903 ... |w-x| <= |x-y| <= |y-z| .......... G
A212904 ... |w-x| + |x-y| + |y-z| = n ........ H
A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H
...
Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.
B: 3,-1,-5,5,1,-3,1
C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1
D: 4,-5,0,5,-4,1
E: 1,3,-3,-3,3,1,-1
F: 1,4,-4,-6,6,4,-4,-1,1
G: 2,1,-3,-1,1,3,-1,-2,1
H: 2,1,-4,1,2,-1

		Examples
					a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).
		
	
References
			A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

		
Links
			Bo Gyu Jeong, Table of n, a(n) for n = 0..200

		
CrossrefsCf. A000583 (n^4), A210000, A211809, A212959.
Programs
			
				Mathematica
				t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},
      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211795 *)
(* Peter J. C. Moses, Apr 13 2012 *)
				

		
Formulaa(n) = n^4 - A211809(n).

A007531 a(n) = n(n-1)(n-2) (or n!/(n-3)!).

Original entry on oeis.org

0, 0, 0, 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, 2184, 2730, 3360, 4080, 4896, 5814, 6840, 7980, 9240, 10626, 12144, 13800, 15600, 17550, 19656, 21924, 24360, 26970, 29760, 32736, 35904, 39270, 42840, 46620, 50616, 54834, 59280, 63960, 68880

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Ed Pegg Jr conjectures that n^3 - n = k! has a solution if and only if n is 2, 3, 5 or 9 (when k is 3, 4, 5 and 6).
Three-dimensional promic (or oblong) numbers, cf. A002378. - Alexandre Wajnberg, Dec 29 2005
Doubled first differences of tritriangular numbers A050534(n) = (1/8)n(n + 1)(n - 1)(n - 2). a(n) = 2*(A050534(n+1) - A050534(n)). - Alexander Adamchuk, Apr 11 2006
If Y is a 4-subset of an n-set X then, for n >= 6, a(n-4) is the number of (n-5)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007
Convolution of A005843 with A008585. - Reinhard Zumkeller, Mar 07 2009
a(n) = A000578(n) - A000567(n). - Reinhard Zumkeller, Sep 18 2009
For n > 3: a(n) = A173333(n, n-3). - Reinhard Zumkeller, Feb 19 2010
Let H be the n X n Hilbert matrix H(i, j) = 1/(i+j-1) for 1 <= i, j <= n. Let B be the inverse matrix of H. The sum of the elements in row 2 of B equals (-1)^n a(n+1). - T. D. Noe, May 01 2011
a(n) equals 2^(n-1) times the coefficient of log(3) in 2F1(n-2, n-2, n, -2). - John M. Campbell, Jul 16 2011
For n > 2 a(n) = 1/(Integral_{x = 0..Pi/2} (sin(x))^5*(cos(x))^(2*n-5)). - Francesco Daddi, Aug 02 2011
a(n) is the number of functions f:[3] -> [n] that are injective since there are n choices for f(1), (n-1) choices for f(2), and (n-2) choices for f(3). Also, a(n+1) is the number of functions f:[3] -> [n] that are width-2 restricted (that is, the pre-image under f of any element in [n] is of size 2 or less). See "Width-restricted finite functions" link below. - Dennis P. Walsh, Mar 01 2012
This sequence is produced by three consecutive triangular numbers t(n-1), t(n-2) and t(n-3) in the expression 2*t(n-1)*(t(n-2)-t(n-3)) for n = 0, 1, 2, ... - J. M. Bergot, May 14 2012
For n > 2: A020639(a(n)) = 2; A006530(a(n)) = A093074(n-1). - Reinhard Zumkeller, Jul 04 2012
Number of contact points between equal spheres arranged in a tetrahedron with n - 1 spheres in each edge. - Ignacio Larrosa Cañestro, Jan 07 2013
Also for n >= 3, area of Pythagorean triangle in which one side differs from hypotenuse by two units. Consider any Pythagorean triple (2n, n^2-1, n^2+1) where n > 1. The area of such a Pythagorean triangle is n(n^2-1). For n = 2, 3, 4,.. the areas are 6, 24, 60, .... which are the given terms of the series. - Jayanta Basu, Apr 11 2013
Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graph K_3. - Tom Copeland, Apr 05 2014
Starting with 6, 24, 60, 120, ..., a(n) is the number of permutations of length n>=3 avoiding the partially ordered pattern (POP) {1>2} of length 5. That is, the number of length n permutations having no subsequences of length 5 in which the first element is larger than the second element. - Sergey Kitaev, Dec 11 2020
For integer m and positive integer r >= 2, 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) = (2 - r*m)/2 in the complex plane. - Peter Bala, Jun 02 2024

R. K. Guy, Unsolved Problems in Theory of Numbers, Section D25.
L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 40.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. Math., 14:42-4. See 1st polynomial p. 5.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Michael Penn, A natural nested root, YouTube video, 2022.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014-2015.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
Dennis Walsh, Width-restricted finite functions
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

binomial(n, k): A161680 (k = 2), A000332 (k = 4), A000389 (k = 5), A000579 (k = 6), A000580 (k = 7), A000581 (k = 8), A000582 (k = 9).
Cf. A002378, A005563, A084939, A084940, A084941, A084942, A084943, A084944.
Cf. A028896.

a007531 n = product [n-2..n]  -- Reinhard Zumkeller, Jul 04 2012

[n*(n-1)*(n-2): n in [0..40]]; // Vincenzo Librandi, May 02 2011

[seq(6*binomial(n,3),n=0..41)]; # Zerinvary Lajos, Nov 24 2006

Table[n^3 - 3n^2 + 2n, {n, 0, 42}]
Table[FactorialPower[n, 3], {n, 0, 42}] (* Arkadiusz Wesolowski, Oct 29 2012 *)

PARI
```
a(n)=n*(n-1)*(n-2)
```

[n*(n-1)*(n-2) for n in range(40)] # G. C. Greubel, Feb 11 2019

a(n) = 6*A000292(n-2).
a(n) = Sum_{i=1..n} polygorial(3,i) where polygorial(3,i) = A028896(i-1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6, n > 2. - Zak Seidov, Feb 09 2006
G.f.: 6*x^2/(1-x)^4.
a(-n) = -a(n+2).
1/6 + 3/24 + 5/60 + ... = Sum_{k>=1} (2*k-1)/(k*(k+1)*(k+2)) = 3/4. [Jolley Eq. 213]
a(n+1) = n^3 - n. - Mohammad K. Azarian, Jul 26 2007
E.g.f.: x^3*exp(x). - Geoffrey Critzer, Feb 08 2009
If the first 0 is eliminated, a(n) = floor(n^5/(n^2+1)). - Gary Detlefs, Feb 11 2010
1/6 + 1/24 + 1/60 + ... = Sum_{n>=1} 1/(n*(n+1)*(n+2)) = 1/4. - Mohammad K. Azarian, Dec 29 2010
a(0) = 0, a(n) = a(n-1) + 3*(n-1)*(n-2). - Jean-François Alcover, Jan 08 2013
(a(n+1) - a(n))/6 = A000217(n-2) for n > 0. - J. M. Bergot, Jul 30 2013
Partial sums of A028896. - R. J. Mathar, Aug 28 2014
1/6 + 1/24 + 1/60 + ... + 1/(n*(n+1)*(n+2)) = n*(n+3)/(4*(n+1)*(n+2)). - Christina Steffan, Jul 20 2015
a(n+2)^2 = A005563(n)^3 + A005563(n)^2. - Bruno Berselli, May 03 2018
a(n)*a(n+1) + A000096(n-3)^2 = m^2 (a perfect square), m = ((a(n)+a(n+1))/2)-n. - Ezhilarasu Velayutham, May 21 2019
Sum_{n>=3} (-1)^(n+1)/a(n) = 2*log(2) - 5/4. - Amiram Eldar, Jul 02 2020
For n >= 3, (a(n) + (a(n) + (a(n) + ...)^(1/3))^(1/3))^(1/3) = n - 1. - Paolo Xausa, Apr 09 2022

A001296 4-dimensional pyramidal numbers: a(n) = (3n+1)binomial(n+2, 3)/4. Also Stirling2(n+2, n).

Original entry on oeis.org

0, 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, 2431, 3367, 4550, 6020, 7820, 9996, 12597, 15675, 19285, 23485, 28336, 33902, 40250, 47450, 55575, 64701, 74907, 86275, 98890, 112840, 128216, 145112, 163625, 183855, 205905, 229881, 255892, 284050, 314470

Offset: 0

N. J. A. Sloane

Permutations avoiding 12-3 that contain the pattern 31-2 exactly once.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Partial sums of A002411. - Jonathan Vos Post, Mar 16 2006
If Y is a 3-subset of an n-set X then, for n>=6, a(n-5) is the number of 6-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
Starting with 1 = binomial transform of [1, 6, 12, 10, 3, 0, 0, 0, ...]. Equals row sums of triangle A143037. - Gary W. Adamson, Jul 18 2008
Rephrasing the Perry formula of 2003: a(n) is the sum of all products of all two  numbers less than or equal to n, including the squares. Example: for n=3 the sum of these products is 1*1 + 1*2 + 1*3 + 2*2 + 2*3 + 3*3 = 25. - J. M. Bergot, Jul 16 2011
Half of the partial sums of A011379. [Jolley, Summation of Series, Dover (1961), page 12 eq (66).] - R. J. Mathar, Oct 03 2011
Also the number of (w,x,y,z) with all terms in {1,...,n+1} and w < x >= y > z (see A211795). - Clark Kimberling, May 19 2012
Convolution of A000027 with A000326. - Bruno Berselli, Dec 06 2012
This sequence is related to A000292 by a(n) = n*A000292(n) - Sum_{i=0..n-1} A000292(i) for n>0. - Bruno Berselli, Nov 23 2017
a(n-2) is the maximum number of intersections made from the perpendicular bisectors of all pair combinations of n points. - Ian Tam, Dec 22 2020

			G.f. = x + 7*x^2 + 25*x^3 + 65*x^4 + 140*x^5 + 266*x^6 + 462*x^7 + 750*x^8 + 1155*x^9 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/3).
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
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].
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
S. Butler and P. Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, page 5.
M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Chanticleer Press, NY, 1950, p. 36.
C. Krishnamachaki, The operator (xD)^n, J. Indian Math. Soc., 15 (1923),3-4. [Annotated scanned copy]
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 p. 29.
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
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
Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a(n)=f(n, 2) where f is given in A034261.
Cf. A000217, A000326, A001297, A001298, A002411, A008277, A008517, A094262.
a(n)= A093560(n+3, 4), (3, 1)-Pascal column.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
Cf. similar sequences listed in A241765 and A254142.
Cf. A000914.

/* A000027 convolved with A000326: */ A000326:=func; [&+[(n-i+1)*A000326(i): i in [0..n]]: n in [0..40]]; // Bruno Berselli, Dec 06 2012

[(3*n+1)*Binomial(n+2,3)/4: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

A001296:=-(1+2*z)/(z-1)**5; # Simon Plouffe in his 1992 dissertation for sequence without the leading zero

Table[n*(1+n)*(2+n)*(1+3*n)/24, {n, 0, 100}]
CoefficientList[Series[x (1 + 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Table[StirlingS2[n+2, n], {n, 0, 40}] (* Jean-François Alcover, Jun 24 2015 *)
Table[ListCorrelate[Accumulate[Range[n]],Range[n]],{n,0,40}]//Flatten (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,7,25,65},40] (* Harvey P. Dale, Aug 14 2017 *)

t(n)=n*(n+1)/2
for(i=1,30,print1(","sum(j=1,i,j*t(j))))

{a(n) = n * (n+1) * (n+2) * (3*n+1) / 24}; /* Michael Somos, Sep 04 2017 */

[stirling_number2(n+2,n) for n in range(0,38)] # Zerinvary Lajos, Mar 14 2009

a(n) = n*(1+n)*(2+n)*(1+3*n)/24. - T. D. Noe, Jan 21 2008
G.f.: x*(1+2*x)/(1-x)^5. - Paul Barry, Jul 23 2003
a(n) = Sum_{j=0..n} j*A000217(j). - Jon Perry, Jul 28 2003
E.g.f. with offset -1: exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!). For the coefficients [1, 4, 3] see triangle A112493.
E.g.f. x*exp(x)*(24 + 60*x + 28*x^2 + 3*x^3)/24 (above e.g.f. differentiated).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3. - Kieren MacMillan, Sep 29 2008
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Jaume Oliver Lafont, Nov 23 2008
O.g.f. is D^2(x/(1-x)) = D^3(x), where D is the operator x/(1-x)*d/dx. - Peter Bala, Jul 02 2012
a(n) = A153978(n)/2. - J. M. Bergot, Aug 09 2013
a(n) = A002817(n) + A000292(n-1). - J. M. Bergot, Aug 29 2013; [corrected by Cyril Damamme, Feb 26 2018]
a(n) = A000914(n+1) - 2 * A000330(n+1). - Antal Pinter, Dec 31 2015
a(n) = A080852(3,n-1). - R. J. Mathar, Jul 28 2016
a(n) = 1*(1+2+...+n) + 2*(2+3+...+n) + ... + n*n. For example, a(6) = 266 = 1(1+2+3+4+5+6) + 2*(2+3+4+5+6) + 3*(3+4+5+6) + 4*(4+5+6) + 5*(5+6) + 6*(6).- J. M. Bergot, Apr 20 2017
a(n) = A000914(-2-n) for all n in Z. - Michael Somos, Sep 04 2017
a(n) = A000292(n) + A050534(n+1). - Cyril Damamme, Feb 26 2018
From Amiram Eldar, Jul 02 2020: (Start)
Sum_{n>=1} 1/a(n) = (6/5) * (47 - 3*sqrt(3)*Pi - 27*log(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = (6/5) * (16*log(2) + 6*sqrt(3)*Pi - 43). (End)

A027480 a(n) = n(n+1)(n+2)/2.

Original entry on oeis.org

0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

Offset: 0

Olivier Gérard and Ken Knowlton (kcknowlton(AT)aol.com)

Write the integers in groups: 0; 1,2; 3,4,5; 6,7,8,9; ... and add the groups: a(n) = Sum_{j=0..n} (A000217(n)+j), row sums of the triangular view of A001477. - Asher Auel, Jan 06 2000
With offset = 2, a(n) is the number of edges of the line graph of the complete graph of order n, L(K_n). - Roberto E. Martinez II, Jan 07 2002
Also the total number of pips on a set of dominoes of type n. (A "3" domino set would have 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, 3-3.) - Gerard Schildberger, Jun 26 2003. See A129533 for generalization to n-armed "dominoes". - N. J. A. Sloane, Jan 06 2016
Common sum in an (n+1) X (n+1) magic square with entries (0..n^2-1).
Alternate terms of A057587. - Jeremy Gardiner, Apr 10 2005
If Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of 4-subsets of X which have exactly one element in common with Y. Also, if Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-5)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
These numbers, starting with 3, are the denominators of the power series f(x) = (1-x)^2 * log(1/(1-x)), if the numerators are kept at 1. This sequence of denominators starts at the term x^3/3. - Miklos Bona, Feb 18 2009
a(n) is the number of triples (w,x,y) having all terms in {0..n} and satisfying at least one of the inequalities x+y < w, y+w < x, w+x < y. - Clark Kimberling, Jun 14 2012
From Martin Licht, Dec 04 2016: (Start)
Let b(n) = (n+1)(n+2)(n+3)/2 (the same sequence, but with a different offset). Then (see Arnold et al., 2006):
b(n) is the dimension of the Nédélec space of the second kind of polynomials of order n over a tetrahedron.
b(n-1) is the dimension of the curl-conforming Nédélec space of the first kind of polynomials of order n with tangential boundary conditions over a tetrahedron.
b(n) is the dimension of the divergence-conforming Nédélec space of the first kind of polynomials of order n with normal boundary conditions over a tetrahedron. (End)
After a(0), the digital root has period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9]. - Peter M. Chema, Jan 19 2017

			Row sums of n consecutive integers, starting at 0, seen as a triangle:
.
    0 |  0
    3 |  1  2
   12 |  3  4  5
   30 |  6  7  8  9
   60 | 10 11 12 13 14
  105 | 15 16 17 18 19 20

T. D. Noe, Table of n, a(n) for n = 0..1000
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15 (2006), 1-155.
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Solomon Gartenhaus, Odd Order Pandiagonal Latin and Magic Cubes in Three and Four Dimensions, arXiv:math/0210275 [math.CO], 2002.
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/beta(n, 3) in A061928.
Cf. A057587, A006003, A254407.
A row of array in A129533.
Cf. similar sequences of the type n*(n+1)*(n+k)/2 listed in A267370.
Similar sequences are listed in A316224.
Cf. A056923, A281258.
Third column of A003506.
Cf. A000217, A002378, A006002.
A bisection of A330298.

[n*(n+1)*(n+2)/2: n in [0..40]]; // Vincenzo Librandi, Nov 14 2014

[seq(3*binomial(n+2,3),n=0..37)]; # Zerinvary Lajos, Nov 24 2006
a := n -> add((j+n)*(n+2)/3,j=0..n): seq(a(n),n=0..35); # Zerinvary Lajos, Dec 17 2006

Table[(m^3 - m)/2, {m, 36}] (* Zerinvary Lajos, Mar 21 2007 *)
LinearRecurrence[{4,-6,4,-1},{0,3,12,30},40] (* Harvey P. Dale, Oct 10 2012 *)
CoefficientList[Series[3 x / (x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 14 2014 *)
With[{nn=50},Total/@TakeList[Range[0,(nn(nn+1))/2-1],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jun 02 2019 *)

a(n)=3*binomial(n+2,3) \\ Charles R Greathouse IV, May 23 2011

def a(n): return (n**3+3*n**2+2*n)//2 # _Torlach Rush, Jun 16 2024

a(n) = a(n-1) + A050534(n) = 3*A000292(n-1) = A050534(n) - A050534(n-1).
a(n) = n*binomial(2+n, 2). - Zerinvary Lajos, Jan 10 2006
a(n) = A007531(n+2)/2. - Zerinvary Lajos, Jul 17 2006
Starting with offset 1 = binomial transform of [3, 9, 9, 3, 0, 0, 0]. - Gary W. Adamson, Oct 25 2007
From R. J. Mathar, Apr 07 2009: (Start)
G.f.: 3*x/(x-1)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
a(n) = Sum_{i=0..n} n*(n - i) + 2*i. - Bruno Berselli, Jan 13 2016
From Ilya Gutkovskiy, Aug 07 2016: (Start)
E.g.f.: x*(6 + 6*x + x^2)*exp(x)/2.
a(n) = Sum_{k=0..n} A045943(k).
Sum_{n>=1} 1/a(n) = 1/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (8*log(2) - 5)/2 = 0.2725887222397812... = A016639/10. (End)
a(n-1) = binomial(n^2,2)/n for n > 0. - Jonathan Sondow, Jan 07 2018
For k > 1, Sum_{i=0..n^2-1} (k+i)^2 = (k*n + a(k-1))^2 + A126275(k). - Charlie Marion, Apr 23 2021

A001498 Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 15, 15, 1, 10, 45, 105, 105, 1, 15, 105, 420, 945, 945, 1, 21, 210, 1260, 4725, 10395, 10395, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 1, 36, 630, 6930, 51975, 270270, 945945, 2027025, 2027025, 1, 45, 990, 13860, 135135, 945945, 4729725, 16216200, 34459425, 34459425

Offset: 0

N. J. A. Sloane

The row polynomials with exponents in increasing order (e.g., third row: 1+3x+3x^2) are Grosswald's y_{n}(x) polynomials, p. 18, Eq. (7).
Also called Bessel numbers of first kind.
The triangle a(n,k) has factorization [C(n,k)][C(k,n-k)]Diag((2n-1)!!) The triangle a(n-k,k) is A100861, which gives coefficients of scaled Hermite polynomials. - Paul Barry, May 21 2005
Related to k-matchings of the complete graph K_n by a(n,k)=A100861(n+k,k). Related to the Morgan-Voyce polynomials by a(n,k)=(2k-1)!!*A085478(n,k). - Paul Barry, Aug 17 2005
Related to Hermite polynomials by a(n,k)=(-1)^k*A060821(n+k, n-k)/2^n. - Paul Barry, Aug 28 2005
The row polynomials, the Bessel polynomials y(n,x):=Sum_{m=0..n} (a(n,m)*x^m) (called y_{n}(x) in the Grosswald reference) satisfy (x^2)*(d^2/dx^2)y(n,x) + 2*(x+1)*(d/dx)y(n,x) - n*(n+1)*y(n,x) = 0.
a(n-1, m-1), n >= m >= 1, enumerates unordered n-vertex forests composed of m plane (aka ordered) increasing (rooted) trees. Proof from the e.g.f. of the first column Y(z):=1-sqrt(1-2*z) (offset 1) and the Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w). See their remark on p. 28 on plane recursive trees. For m=1 see the D. Callan comment on A001147 from Oct 26 2006. - Wolfdieter Lang, Sep 14 2007
The asymptotic expansions of the higher order exponential integrals E(x,m,n), see A163931 for information, lead to the Bessel numbers of the first kind in an intriguing way. For the first four values of m these asymptotic expansions lead to the triangles A130534 (m=1), A028421 (m=2), A163932 (m=3) and A163934 (m=4). The o.g.f.s. of the right hand columns of these triangles in their turn lead to the triangles A163936 (m=1), A163937 (m=2), A163938 (m=3) and A163939 (m=4). The row sums of these four triangles lead to A001147, A001147 (minus a(0)), A001879 and A000457 which are the first four right hand columns of A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to A001880, m=6 to A001881, m=7 to A038121 and m=8 to A130563 which are the next four right hand columns of A001498. So one by one all columns of the triangle of coefficients of Bessel polynomials appear. - Johannes W. Meijer, Oct 07 2009
a(n,k) also appear as coefficients of (n+1)st degree of the differential operator D:=1/t d/dt, namely D^{n+1}= Sum_{k=0..n} a(n,k) (-1)^{n-k} t^{1-(n+k)} (d^{n+1-k}/dt^{n+1-k}. - Leonid Bedratyuk, Aug 06 2010
a(n-1,k) are the coefficients when expanding (xI)^n in terms of powers of I. Let I(f)(x) := Integral_{a..x} f(t) dt, and (xI)^n := x Integral_{a..x} [ x_{n-1} Integral_{a..x_{n-1}} [ x_{n-2} Integral_{a..x_{n-2}} ... [ x_1 Integral_{a..x_1} f(t) dt ] dx_1 ] .. dx_{n-2} ] dx_{n-1}. Then: (xI)^n = Sum_{k=0..n-1} (-1)^k * a(n-1,k) * x^(n-k) * I^(n+k)(f)(x) where I^(n) denotes iterated integration. - Abdelhay Benmoussa, Apr 11 2025

			The triangle a(n, k), n >= 0, k = 0..n, begins:
  1
  1  1
  1  3   3
  1  6  15    15
  1 10  45   105    105
  1 15 105   420    945    945
  1 21 210  1260   4725  10395   10395
  1 28 378  3150  17325  62370  135135   135135
  1 36 630  6930  51975 270270  945945  2027025  2027025
  1 45 990 13860 135135 945945 4729725 16216200 34459425 34459425
  ...
And the first few Bessel polynomials are:
  y_0(x) = 1,
  y_1(x) = x + 1,
  y_2(x) = 3*x^2 + 3*x + 1,
  y_3(x) = 15*x^3 + 15*x^2 + 6*x + 1,
  y_4(x) = 105*x^4 + 105*x^3 + 45*x^2 + 10*x + 1,
  y_5(x) = 945*x^5 + 945*x^4 + 420*x^3 + 105*x^2 + 15*x + 1,
  ...
Tree counting: a(2,1)=3 for the unordered forest of m=2 plane increasing trees with n=3 vertices, namely one tree with one vertex (root) and another tree with two vertices (a root and a leaf), labeled increasingly as (1, 23), (2,13) and (3,12). - _Wolfdieter Lang_, Sep 14 2007

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

T. D. Noe, Rows n=0..50 of triangle, flattened
Alexander Alldridge, Joachim Hilgert, and Martin R. Zirnbauer, Chevalley's restriction theorem for reductive symmetric superpairs, arXiv:0812.3530 [math.RT], 2008-2009; J. Alg. 323 (4) (2010) 1159-1185 doi:10.1016/j.jalgebra.2009.11.014, Remark 3.17.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
Juan Antonio Barcelo and Anthony Carbery, On the magnitudes of compact sets in Euclidean spaces, arXiv preprint arXiv:1507.02502 [math.MG], 2015.
François Bergeron, Philippe Flajolet, and Bruno Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
Alexander W. Boldyreff, Decomposition of Rational Fractions into Partial Fractions, Nat. Math. Mag. 17 (6) (1943), 261-267; coefficients (m)N(r).
Alexander Burstein and Toufik Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.
Roudy El Haddad, Repeated Integration and Explicit Formula for the n-th Integral of x^m*(ln x)^m', arXiv:2102.11723 [math.GM], 2021.
Andrew Francis and Michael Hendriksen, Counting spinal phylogenetic networks, arXiv:2502.14223 [q-bio.PE], 2025. See p. 9.
Emil Grosswald, Bessel Polynomials: Recurrence Relations, Lecture Notes Math. vol. 698, 1978, p. 18.
Cameron Jakub and Mihai Nica, Depth Degeneracy in Neural Networks: Vanishing Angles in Fully Connected ReLU Networks on Initialization, arXiv:2302.09712 [stat.ML], 2023.
Taekyun Kim, and Dae San Kim, Identities involving Bessel polynomials arising from linear differential equations, arXiv:1602.04106 [math.NT], 2016.
H. L. Krall and Orrin Frink, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First ten rows.
B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
Shi-Mei Ma, Toufik Mansour, and Matthias Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Shi-Mei Ma, Toufik Mansour, Jean Yeh, and Yeong-Nan Yeh, Normal ordered grammars, arXiv:2404.15119 [math.CO], 2024. See p. 11.
Guillermo Navas-Palencia, On the computation of the cumulative distribution function of the Normal Inverse Gaussian distribution, arXiv:2502.16015 [math.NA], 2025. See p. 25.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
John Riordan, Notes to N. J. A. Sloane, Jul. 1968
Florian Stober, Average case considerations for MergeInsertion, Master's Thesis, University of Stuttgart, Institute of Formal Methods in Computer Science, 2018.
Florian Stober and Armin Weiß, On the Average Case of MergeInsertion, arXiv:1905.09656 [cs.DS], 2019.
Laszlo A. Székely, Pál L. Erdős, and M. A. Steel, The combinatorics of evolutionary trees, Séminaire Lotharingien de Combinatoire, B28e (1992), 15 pp.
Juan G. Triana, Bessel polynomials by context-free grammars (Polinomios de Bessel mediante gramáticas independientes del contexto), Bistua, Univ. de Pamplona (Colombia, 2024) Vol 22, No. 2. See p. 3.
Jonas Wahl, Traces on diagram algebras II: Centralizer algebras of easy groups and new variations of the Young graph, arXiv:2009.08181 [math.RT], 2020.
Eric Weisstein's World of Mathematics, Modified Spherical Bessel Function of the Second Kind
Index entries for sequences related to Bessel functions or polynomials

Cf. A001497 (same triangle but rows read in reverse order). Other versions of this same triangle are given in A144331, A144299, A111924 and A100861.
Columns from left edge include A000217, A050534.
Columns 1-6 from right edge are A001147, A001879, A000457, A001880, A001881, A038121.
Bessel polynomials evaluated at certain x are A001515 (x=1, row sums), A000806 (x=-1), A001517 (x=2), A002119 (x=-2), A001518 (x=3), A065923 (x=-3), A065919 (x=4). Cf. A043301, A003215.
Cf. A245066 (central terms). A113025 (y_n(2*x)).

a001498 n k = a001498_tabl !! n !! k
a001498_row n = a001498_tabl !! n
a001498_tabl = map reverse a001497_tabl
-- Reinhard Zumkeller, Jul 11 2014

/* As triangle: */ [[Factorial(n+k)/(2^k*Factorial(n-k)*Factorial(k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 15 2016

Bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end; # explicit Bessel polynomials
Bessel := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*Bessel(n-1)+Bessel(n-2); fi; end; # recurrence for Bessel polynomials
bessel := proc(n,x) add(binomial(n+k,2*k)*(2*k)!*x^k/(k!*2^k),k=0..n); end;
f := proc(n) option remember; if n <=1 then (1+x)^n else (2*n-1)*x*f(n-1)+f(n-2); fi; end;
# Alternative:
T := (n,k) -> pochhammer(n+1,k)*binomial(n,k)/2^k:
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 11 2018
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (n - k + 1)* T(n, k - 1) + T(n - 1, k) fi fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Oct 02 2023

max=50; Flatten[Table[(n+k)!/(2^k*(n-k)!*k!), {n, 0, Sqrt[2 max]//Ceiling}, {k, 0, n}]][[1 ;; max]] (* Jean-François Alcover, Mar 20 2011 *)

{T(n,k)=if(k<0||k>n, 0, binomial(n, k)*(n+k)!/2^k/n!)} /* Michael Somos, Oct 03 2006 */

A001497_ser(N,t='t) = {
  my(x='x+O('x^(N+2)));
  serlaplace(deriv(exp((1-sqrt(1-2*t*x))/t),'x));
};
concat(apply(Vecrev, Vec(A001497_ser(9)))) \\ Gheorghe Coserea, Dec 27 2017

a(n, k) = (n+k)!/(2^k*(n-k)!*k!) (see Grosswald and Riordan). - Ralf Stephan, Apr 20 2004
a(n, 0)=1; a(0, k)=0, k > 0; a(n, k) = a(n-1, k) + (n-k+1) * a(n, k-1) = a(n-1, k) + (n+k-1) * a(n-1, k-1). - Len Smiley
a(n, m) = A001497(n, n-m) = A001147(m)*binomial(n+m, 2*m) for n >= m >= 0, otherwise 0.
G.f. for m-th column: (A001147(m)*x^m)/(1-x)^(2*m+1), m >= 0, where A001147(m) = double factorials (from explicit a(n, m) form).
Row polynomials y_n(x) are given by D^(n+1)(exp(t)) evaluated at t = 0, where D is the operator 1/(1-t*x)*d/dt. - Peter Bala, Nov 25 2011
G.f.: conjecture: T(0)/(1-x), where T(k) = 1 - x*y*(k+1)/(x*y*(k+1) - (1-x)^2/T(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013
Recurrence from Grosswald, p. 18, eq. (5), for the row polynomials: y_n(x) = (2*n-1)*x*y_{n-1} + y_{n-2}(x), y_{-1}(x) = 1 = y_{0} = 1, n >= 1. This becomes, for n >= 0, k = 0..n: a(n, k) = 0 for n < k (zeros not shown in the triangle), a(n, -1) = 0, a(0, 0) = 1 = a(1, 0) and otherwise a(n, k) = (2*n-1)*a(n-1, k-1) + a(n-2, k). Compare with the above given recurrences. - Wolfdieter Lang, May 11 2018
T(n, k) = Pochhammer(n+1,k)*binomial(n,k)/2^k = A113025(n,k)/2^k. - Peter Luschny, May 11 2018
a(n, k) = Sum_{i=0..min(n-1, k)} (n-i)(k-i) * a(n-1, i) where x(n) = x*(x-1)*...*(x-n+1) is the falling factorial, this equality follows directly from the operational formula we wrote in Apr 11 2025.- Abdelhay Benmoussa, May 18 2025

A034827 a(n) = 2*binomial(n,4).

Original entry on oeis.org

0, 0, 0, 0, 2, 10, 30, 70, 140, 252, 420, 660, 990, 1430, 2002, 2730, 3640, 4760, 6120, 7752, 9690, 11970, 14630, 17710, 21252, 25300, 29900, 35100, 40950, 47502, 54810, 62930, 71920, 81840, 92752, 104720, 117810, 132090, 147630, 164502, 182780

Offset: 0

N. J. A. Sloane

Also number of ways to insert two pairs of parentheses into a string of n-4 letters (allowing empty pairs of parentheses). E.g., there are 30 ways for 2 letters. Cf. A002415.
2,10,30,70, ... gives orchard crossing number of complete graph K_n. - Ralf Stephan, Mar 28 2003
If Y is a 2-subset of an n-set X then, for n>=4, a(n-1) is the number of 4-subsets and 5-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Middle column of table on p. 6 of Feder and Garber. - Jonathan Vos Post, Apr 23 2009
Number of pairs of non-intersecting lines when each of n points around a circle is joined to every other point by straight lines. A pair of lines is considered non-intersecting if the lines do not intersect in either the interior or the boundary of a circle. - Melvin Peralta, Feb 05 2016
From a(2), convolution of the oblong numbers (A002378) with the nonnegative numbers (A001477). - Bruno Berselli, Oct 24 2016
Also the number of 3-cycles in the n-triangular honeycomb bishop graph. - Eric W. Weisstein, Aug 10 2017

Charles Jordan, Calculus of Finite Differences, Chelsea, 1965, p. 449.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Bruno Berselli, Table of n, a(n) for n = 0..1000
M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, arXiv:hep-th/0105045, 2001.
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
Elie Feder and David Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003-2009.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A diagonal of A088617.
Cf. A033487, A050534, A060008.
Partial sums of A007290.
Cf. A001477, A002378.
Cf. A051843 (4-cycles in the triangular honeycomb bishop graph), A290775 (5-cycles), A290779 (6-cycles).

[2*Binomial(n,4): n in [0..40]]; // Vincenzo Librandi, Oct 20 2013

[seq(binomial(n,4)*2,n=0..40)]; # Zerinvary Lajos, Jul 18 2006

CoefficientList[Series[2 x^4/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 20 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 0, 2}, 50] (* Harvey P. Dale, Jun 09 2016 *)
Table[2 Binomial[n, 4], {n, 0, 40}] (* Bruno Berselli, Oct 24 2016 *)
2 Binomial[Range[0, 20], 4] (* Eric W. Weisstein, Aug 10 2017 *)

a(n)=2*binomial(n,4) \\ Charles R Greathouse IV, Jun 23 2015

a(n) = A096338(2*n-6) = 2*A000332(n), n>2. - R. J. Mathar, Nov 08 2010
G.f.: 2*x^4/(1-x)^5. - Colin Barker, Feb 29 2012
a(n) = Sum_{k=1..n-3} ( Sum_{i=1..k} i*(2*k-n+4) ). - Wesley Ivan Hurt, Sep 26 2013
E.g.f.: x^4*exp(x)/12. - G. C. Greubel, Feb 23 2017
From Amiram Eldar, Jul 19 2022: (Start)
Sum_{n>=4} 1/a(n) = 2/3.
Sum_{n>=4} (-1)^n/a(n) = 16*log(2) - 32/3. (End)

A033487 a(n) = n(n+1)(n+2)*(n+3)/4.

Original entry on oeis.org

0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, 6006, 8190, 10920, 14280, 18360, 23256, 29070, 35910, 43890, 53130, 63756, 75900, 89700, 105300, 122850, 142506, 164430, 188790, 215760, 245520, 278256, 314160, 353430, 396270, 442890, 493506, 548340, 607620

Offset: 0

N. J. A. Sloane

Non-vanishing diagonal of (A132440)^4/4. Third subdiagonal of unsigned A238363 without the zero. Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices of the complete graph K_4. - Tom Copeland, Apr 05 2014
Total number of pips on a set of trominoes (3-armed dominoes) with up to n pips on each arm. - Alan Shore and N. J. A. Sloane, Jan 06 2016
Also the number of minimum connected dominating sets in the (n+2)-crown graph. - Eric W. Weisstein, Jun 29 2017
Crossing number of the (n+3)-cocktail party graph (conjectured). - Eric W. Weisstein, Apr 29 2019
Sum of all numbers in ordered triples (x,y,z) where 0 <= x <= y <= z <= n. - Edward Krogius, Jul 31 2022

			G.f. = 6*x + 30*x^2 + 90*x^3 + 210*x^4 + 420*x^5 + 756*x^6 + 1260*x^7 + ...

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Vincenzo Librandi, Table of n, a(n) for n = 0..690
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq., Vol. 13 (2010), Article 10.4.4.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Aleksandar Petojević and Nenad Đapić, The vAm(a,b,c;z) function, Preprint 2013.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Hasan Unal, Proof Without Words: Sums of Products of Three Consecutive Integers, Mathematics Magazine, Vol. 88, No. 1 (February 2015), pp. 37-38.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Crown Party Graph.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Index entries for sequences related to Bessel functions or polynomials
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000332, A002378, A033486, A033488, A034827, A050534, A130534, A132440, A238363.
Partial sums of A007531.
A row of the array in A129533.
A column of the triangle in A331430.
Sequences of the form binomial(n+k,k)*binomial(n+k+2,k): A000012 (k=0), A005563 (k=1), this sequence (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).

[n*(n+1)*(n+2)*(n+3)/4: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

[seq(binomial(n+3,4)*6, n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Table[Times @@ (n + Range[0, 3])/4, {n, 0, 40}] (* Harvey P. Dale, Nov 27 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 6, 30, 90, 210}, 40] (* Harvey P. Dale, Nov 27 2013 *)
Table[6 Binomial[n+3, 4], {n,0,40}] (* Eric W. Weisstein, Jun 29 2017 *)
Times @@@ Table[n+k, {n, 0, 40}, {k, 0, 3}]/4 (* Eric W. Weisstein, Apr 29 2019 *)

a(n)=6*binomial(n+3,4) \\ Charles R Greathouse IV, Apr 17 2012

concat(0, Vec(6*x/(1-x)^5 + O(x^100))) \\ Altug Alkan, Nov 29 2015

def A033487(n): return 6*binomial(n+3,4)
print([A033487(n) for n in range(41)]) # G. C. Greubel, Feb 08 2025

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001: (Start)
G.f.: 6*x/(1-x)^5.
a(n) = 6*binomial(n+3, 4) = 6*A000332(n+3).
a(n) = a(n-1) + A007531(n+1).
a(n) = Sum_{i=0..n} i*(i+1)*(i+2). (End)
Constant term in Bessel polynomial {y_n(x)}''.
a(n) = binomial(n+1,2)*binomial(n+3,2) = A000217(n)*A000217(n+2). - Zerinvary Lajos, May 25 2005
a(n) = binomial(n+2,2)^2 - binomial(n+2,2). - Zerinvary Lajos, May 17 2006
From Zerinvary Lajos, May 11 2007: (Start)
a(n-1) = Sum_{j=1..n} Sum_{i=2..n} i*j.
a(n) = Sum_{j=1..n} j*(n+2)*(n-1)/2. (End)
Sum_{n>0} 1/a(n) = 2/9. - Enrique Pérez Herrero, Nov 10 2013
a(-3-n) = a(n) = 2 * binomial(binomial(n+2, 2), 2). - Michael Somos, Apr 06 2014
a(n) = A002378(binomial(n+2,2)-1). - Salvador Cerdá, Nov 04 2016
a(n) = Sum_{k=0..n} A007531(k+2). See Proof Without Words link. - Michel Marcus, Oct 29 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 32/9. - Amiram Eldar, Nov 02 2021
E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/4. - Stefano Spezia, Jul 03 2025

A033488 a(n) = n(n+1)(n+2)*(n+3)/6.

Original entry on oeis.org

0, 4, 20, 60, 140, 280, 504, 840, 1320, 1980, 2860, 4004, 5460, 7280, 9520, 12240, 15504, 19380, 23940, 29260, 35420, 42504, 50600, 59800, 70200, 81900, 95004, 109620, 125860, 143840, 163680, 185504, 209440, 235620, 264180, 295260, 329004, 365560, 405080

Offset: 0

N. J. A. Sloane

With two initial 0, convolution of the oblong numbers (A002378) with the nonnegative even numbers (A005843). - Bruno Berselli, Oct 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..700

1/beta(n, 4) in A061928.
Cf. A000332, A034827, A050534.
Cf. A002378, A005843.
Convolution of the oblong numbers with the odd numbers: A008911.
Fourth column of A003506.

[n*(n+1)*(n+2)*(n+3)/6: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

[seq(4*binomial(n+3, 4), n=0..35)]; # Zerinvary Lajos, Nov 24 2006

f[n_]:=n*(n+1)*(n+2)*(n+3)/6; lst={};Do[AppendTo[lst,f[n]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
# Binomial[#+3,3]&/@ Range[0,40]  (* Harvey P. Dale, Feb 20 2011 *)

A033488(n):=n*(n+1)*(n+2)*(n+3)/6$ makelist(A033488(n),n,0,20); /* Martin Ettl, Jan 22 2013 */

a(n) = n*C(3+n, 3). - Zerinvary Lajos, Jan 10 2006
G.f.: 4*x/(1-x)^5. - Colin Barker, Mar 01 2012
G.f.: (2*x/(1-x))*W(0), where W(k) = 1 + 1/( 1 - x*(k+2)*(k+4)/( x*(k+2)*(k+4) + (k+1)*(k+2)/W(k+1) ) ); (continued fraction). - Sergei N. Gladkovskii, Aug 24 2013
From Amiram Eldar, Jun 02 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(2) - 16/3. (End)
E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/6. - Stefano Spezia, Jul 11 2025

A105636 Transform of n^3 by the Riordan array (1/(1-x^2), x).

Original entry on oeis.org

0, 1, 8, 28, 72, 153, 288, 496, 800, 1225, 1800, 2556, 3528, 4753, 6272, 8128, 10368, 13041, 16200, 19900, 24200, 29161, 34848, 41328, 48672, 56953, 66248, 76636, 88200, 101025, 115200, 130816, 147968, 166753, 187272, 209628, 233928, 260281, 288800, 319600

Offset: 0

Paul Barry, Apr 16 2005

Recurrence a(n) = a(n-2) + n^3, starting with a(0)=0, a(1)=1. Also, in physics, a(n)/4 is the trace of the spin operator |S_z|^3 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and therefore the sum of the absolute values of their 3rd powers is 2*28/8 = a(3)/4. - Stanislav Sykora, Nov 07 2013
Also the number of 3-cycles in the (n+1)-triangular honeycomb queen graph. - Eric W. Weisstein, Jul 14 2017
With zero prepended and offset 1, the sequence starts 0,0,1,8,28,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q). Then b(n) is the total volume of the family of cubes with side length |q - p|. - Wesley Ivan Hurt, Apr 14 2018

Stanislav Sykora, Table of n, a(n) for n = 0..9999
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A002620, A000292, A000578, A011934, A231303.

Cf. A289705 (4-cycles), A289706 (5-cycles), A289707 (6-cycles).

List([0..30], n -> (2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16); # G. C. Greubel, Dec 16 2018

[(2*n^4+8*n^3+8*n^2-1)/16+(-1)^n/16: n in [0..50]]; // Vincenzo Librandi, Oct 27 2014

LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 1, 8, 28, 72, 153}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
CoefficientList[Series[x (1 + 4 x + x^2)/((1 + x) (1 - x)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 26 2012 *)
Table[((-1)^n + 2 n^2 (n + 2)^2 - 1)/16, {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)

my(x='x+O('x^99)); concat(0, Vec(x*(1+4*x+x^2)/((1+x)*(1-x)^5))) \\ Altug Alkan, Apr 16 2018

[(2*n^4 +8*n^3 +8*n^2 -1+(-1)^n)/16 for n in range(30)] # G. C. Greubel, Dec 16 2018

G.f.: x*(1+4*x+x^2)/((1+x)*(1-x)^5).
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
a(n) = (2*n^4 + 8*n^3 + 8*n^2 - 1 + (-1)^n)/16.
a(n) = Sum_{k=0..floor((n-1)/2)} (n-2*k)^3.
a(n+1) = Sum_{k=0..n} k^3*(1 - (-1)^(n+k-1))/2.
a(n) = ((((x^2 - (x mod 2) - 4)/4)^2 - (((x^2 - (x mod 2) - 4)/4) mod 2))/8) = floor(((floor(x^2/4) - 1)^2)/8) where x = 2*n + 2. Replace x with 2*n - 1 to obtain A050534(n) = 3*A000332(n+1). Note that a(2*n) = A060300(n)/2 and a(2*n + 1) = A002593(n+1). - Raphie Frank, Jan 30 2014
a(n) = floor(1/(exp(2/n^2) - 1)^2)/2. Also a(n) = A007590(n+1)*A074148(n-1)/2. - Richard R. Forberg, Oct 26 2014
Sum_{n>=1} 1/a(n) = -cot(Pi/sqrt(2))*Pi/sqrt(2) - 1/2. - Amiram Eldar, Aug 25 2022

Showing 1-10 of 54 results. Next

cp's OEIS Frontend

A000292 Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.

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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.

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A007531 a(n) = n*(n-1)*(n-2) (or n!/(n-3)!).

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A001296 4-dimensional pyramidal numbers: a(n) = (3*n+1)*binomial(n+2, 3)/4. Also Stirling2(n+2, n).

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A027480 a(n) = n*(n+1)*(n+2)/2.

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A000292 Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n(n+1)(n+2)/6.

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.

A007531 a(n) = n(n-1)(n-2) (or n!/(n-3)!).

A001296 4-dimensional pyramidal numbers: a(n) = (3n+1)binomial(n+2, 3)/4. Also Stirling2(n+2, n).

A027480 a(n) = n(n+1)(n+2)/2.

A033487 a(n) = n(n+1)(n+2)*(n+3)/4.

A033488 a(n) = n(n+1)(n+2)*(n+3)/6.