A002721 -id:A002721 - cp's OEIS frontend

A002817 Doubly triangular numbers: a(n) = n(n+1)(n^2+n+2)/8.

0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, 3081, 4186, 5565, 7260, 9316, 11781, 14706, 18145, 22155, 26796, 32131, 38226, 45150, 52975, 61776, 71631, 82621, 94830, 108345, 123256, 139656, 157641, 177310, 198765, 222111, 247456, 274911, 304590

Offset: 0

N. J. A. Sloane

Number of inequivalent ways to color vertices of a square using <= n colors, allowing rotations and reflections. Group is dihedral group D_8 of order 8 with cycle index (1/8)*(x1^4 + 2*x4 + 3*x2^2 + 2*x1^2*x2); setting all x_i = n gives the formula a(n) = (1/8)*(n^4 + 2*n + 3*n^2 + 2*n^3).
Number of semi-magic 3 X 3 squares with a line sum of n-1. That is, 3 X 3 matrices of nonnegative integers such that row sums and column sums are all equal to n-1. - [Gupta, 1968, page 653; Bell, 1970, page 279]. - Peter Bertok (peter(AT)bertok.com), Jan 12 2002. See A005045 for another version.
Also the coefficient h_2 of x^{n-3} in the shelling polynomial h(x)=h_0*x^n-1 + h_1*x^n-2 + h_2*x^n-3 + ... + h_n-1 for the independence complex of the cycle matroid of the complete graph K_n on n vertices (n>=2) - Woong Kook (andrewk(AT)math.uri.edu), Nov 01 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-4) is equal to the number of 5-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Starting with offset 1 = binomial transform of [1, 5, 10, 9, 3, 0, 0, 0, ...]. - Gary W. Adamson, Aug 05 2009
Starting with "1" = row sums of triangle A178238. - Gary W. Adamson, May 23 2010
The equation n*(n+1)*(n^2 + n + 2)/8 may be arrived at by solving for x in the following equality: (n^2+n)/2 = (sqrt(8x+1)-1)/2. - William A. Tedeschi, Aug 18 2010
Partial sums of A006003. - Jeremy Gardiner, Jun 23 2013
Doubly triangular numbers are revealed in the sums of row sums of Floyd's triangle.
1, 1+5, 1+5+15, ...
             1
          2     3
       4     5     6
    7     8     9     10
11    12    13    14    15
- Tony Foster III, Nov 14 2015
From Jaroslav Krizek, Mar 04 2017: (Start)
For n>=1; a(n) = sum of the different sums of elements of all the nonempty subsets of the sets of numbers from 1 to n.
Example: for n = 6; nonempty subsets of the set of numbers from 1 to 3: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}; sums of elements of these subsets: 1, 2, 3, 3, 4, 5, 6; different sums of elements of these subsets: 1, 2, 3, 4, 5, 6; a(3) = (1+2+3+4+5+6) = 21, ... (End)
a(n) is also the number of 4-cycles in the (n+4)-path complement graph. - Eric W. Weisstein, Apr 11 2018

			G.f. = x + 6*x^2 + 21*x^3 + 55*x^4 + 120*x^5 + 231*x^6 + 406*x^7 + 666*x^8 + ...

A. Björner, The homology and shellability of matroids and geometric lattices, in Matroid Applications (ed. N. White), Encyclopedia of Mathematics and Its Applications, 40, Cambridge Univ. Press 1992.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(3,r).
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).
R. P. Stanley, Enumerative Combinatorics I, p. 292.

T. D. Noe and William A. Tedeschi, Table of n, a(n) for n = 0..10000 (first 1000 terms computed by T. D. Noe)
G. E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon's partition analysis V. Bijections, recursions and magic squares, p. 37.
Weymar Astaiza, Alexander J. Barrios, Henry Chimal-Dzul, Stephan Ramon Garcia, Jaaziel de la Luz, Victor H. Moll, Yunied Puig, and Diego Villamizar, Symmetric tensor powers of graphs, arXiv:2309.13741 [math.CO], 2023. See p. 12.
Matthias Beck, The number of "magic" squares and hypercubes, arXiv:math/0201013 [math.CO], 2002-2005.
A. G. Bell, Partitioning integers in n dimensions, The Computer Journal, 13 (1970), 278-283.
Miklos Bona, A New Proof of the Formula for the Number of 3 X 3 Magic Squares, Mathematics Magazine, Vol. 70, No. 3 (Jun., 1997), pp. 201-203.
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33(4) (1966), pp. 771-782.
Brian Conrey and Alex Gamburd, Pseudomoments of the Riemann zeta-function and pseudomagic squares, Journal of Number Theory, Volume 117, Issue 2, April 2006, Pages 263-278. See H4 on p. 269.
P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R2.
Robert W. Donley, Partitions for semi-magic squares of size three, arXiv:1911.00977 [math.CO], 2019.
I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
I. J. Good, On the application of symmetric Dirichlet distributions and contingency tables, pp. 1178-1179. (Annotated scanned copy)
Hansraj Gupta, Enumeration of symmetric matrices, Duke Math. J. 35 (3), 653-659, (September 1968).
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
Milan Janjic, Two Enumerative Functions
Neven Juric, Illustration of the 55 3 X 3 matrices
Michal Opler, Pavel Valtr, and Tung Anh Vu, On the Arrangement of Hyperplanes Determined by n Points, EuroCG (39th European Workshop on Computational Geometry, Barcelona, Spain 2023) Session 7B, Talk 1, Vol. 54, No. 6.
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
Henry Warburton, On Self-Repeating Series, Transactions of the Cambridge Philosophical Society, Vol. 9, 471-486, 1856.
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Path Complement Graph
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A006003 (first differences), A165211 (mod 2).
Multiple triangular: A000217, A064322, A066370.
Cf. A001496, A001621, A178238, A005045, A002721.
Cf. A006528 (square colorings).
Cf. A236770 (see crossrefs).
Cf. A016754, A007204.
Row n=3 of A257493 and row n=2 of A331436 and A343097.
Cf. A000332.
Cf. A000292 (3-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}).

Maple
```
A002817 := n->n*(n+1)*(n^2+n+2)/8;
```

a[ n_] := n (n + 1) (n^2 + n + 2) / 8; (* Michael Somos, Jul 24 2002 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,1,6,21,55},40] (* Harvey P. Dale, Jul 18 2011 *)
nn=50;Join[{0},With[{c=(n(n+1))/2},Flatten[Table[Take[Accumulate[Range[ (nn(nn+1))/2]], {c,c}],{n,nn}]]]] (* Harvey P. Dale, Mar 19 2013 *)

{a(n) = n * (n+1) * (n^2 + n + 2) / 8}; /* Michael Somos, Jul 24 2002 */

concat(0, Vec(x*(1+x+x^2)/(1-x)^5 + O(x^50))) \\ Altug Alkan, Nov 15 2015

def A002817(n): return (m:=n*(n+1))*(m+2)>>3 # Chai Wah Wu, Aug 30 2024

a(n) = 3*binomial(n+2, 4) + binomial(n+1, 2).
G.f.: x*(1 + x + x^2)/(1-x)^5. - Simon Plouffe (in his 1992 dissertation); edited by N. J. A. Sloane, May 13 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3. - Warut Roonguthai, Dec 13 1999
a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + a(n-5) = A000217(A000217(n)). - Ant King, Nov 18 2010
a(n) = Sum(Sum(1 + Sum(3*n))). - Xavier Acloque, Jan 21 2003
a(n) = A000332(n+1) + A000332(n+2) + A000332(n+3), with A000332(n) = binomial(n, 4). - Mitch Harris, Oct 17 2006 and Bruce J. Nicholson, Oct 22 2017
a(n) = Sum_{i=1..C(n,2)} i = C(C(n,2) + 1, 2) = A000217(A000217(n+1)). - Enrique Pérez Herrero, Jun 11 2012
Euler transform of length 3 sequence [6, 0, -1]. - Michael Somos, Nov 19 2015
E.g.f.: x*(8 + 16*x + 8*x^2 + x^3)*exp(x)/8. - Ilya Gutkovskiy, Apr 26 2016
Sum_{n>=1} 1/a(n) = 6 - 4*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) = 1.25269064911978447... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(n)*A000124(n)/2.
a(n) = ((n-1)^4 + 3*(n-1)^3 + 2*(n-1)^2 + 2*n))/8. - Bruce J. Nicholson, Apr 05 2017
a(n) = (A016754(n)+ A007204(n)- 2) / 32. - Bruce J. Nicholson, Apr 14 2017
a(n) = a(-1-n) for all n in Z. - Michael Somos, Apr 17 2017
a(n) = T(T(n)) where T are the triangular numbers A000217. - Albert Renshaw, Jan 05 2020
a(n) = 2*n^2 - n + 6*binomial(n, 3) + 3*binomial(n, 4). - Ryan Jean, Mar 20 2021
a(n) = (A008514(n) - 1)/16. - Charlie Marion, Dec 20 2024

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

A006325 4-dimensional analog of centered polygonal numbers.

Original entry on oeis.org

0, 0, 1, 7, 26, 70, 155, 301, 532, 876, 1365, 2035, 2926, 4082, 5551, 7385, 9640, 12376, 15657, 19551, 24130, 29470, 35651, 42757, 50876, 60100, 70525, 82251, 95382, 110026, 126295, 144305, 164176, 186032, 210001, 236215, 264810, 295926

Offset: 0

Albert Rich (Albert_Rich(AT)msn.com)

If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
For n>0, a(n+1) is the n-th antidiagonal sum of A213751. - Clark Kimberling, Jun 20 2012
This sequence is the case m=n-1, k=n+3 of b(m,k) = m*(m+1)*((k-2)*m-(k-5))/6, which is the m-th k-gonal pyramidal number. - Luciano Ancora, Apr 11 2015
Starting (1, 7, 26, ...), this is the binomial transform of (1, 6, 13, 12, 4, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015
Also starting (1, 7, 26, ...), this appears to be the number of magic labelings of the cycle-of-loops graph LOOP X C_4 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. - David J. Seal, Sep 13 2017
The conjecture by David J. Seal is true and easily proved using MacMahon's Omega operators via the "Omega" package for Mathematica authored by Axel Riese (obtaining (up to an offset) the generating function listed in the formula section below). See the second Mathematica program in which the edges of LOOP X C_4 are indexed as in the example below. The Omega package can be downloaded from the link provided in the article by G. E. Andrews et al. - L. Edson Jeffery, Oct 15 2017

			A representation of the LOOP X C_4 graph, with edges and loops indexed as shown, as used in the second Mathematica program below:
.             3         1
.              O_______O
.              |   2   |
.              |4     0|
.              |_______|
.              O   6   O
.             5         7

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
David Galvin and Courtney Sharpe, Independent set sequence of linear hyperpaths, arXiv:2409.15555 [math.CO], 2024. See p. 7.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
J. K. Merikoski, R. Kumar and R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
Richard P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005900, A169938, A213751, A288486, A019298, A061907, A244497, A244879, A244873, A244880.
Cf. A000027, A000217, A019298, A244497, A244879, A244873, A244880, A293310, A293309 (magic labelings of LOOP X C_k, for k = 1..3,5..10).
Cf. A293311, A293312.

[n*(n-1)*(n^2-n+1)/6: n in [0..40]]; // Vincenzo Librandi, May 22 2011

Table[n*(n-1)*(n^2-n+1)/6, {n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
<< Omega.m; n = 4; cond = {}; Do[AppendTo[cond, Sum[a[Mod[2*k - j, 2*n]], {j, 0, 2}] == a[2*n]], {k, 0, n - 1}]; f = OEqSum[Product[x[i]^a[i], {i, 0, 2*n}], cond, u][[1]]; Do[f = OEqR[f, Subscript[u, k]], {k, n}];
(* Generating function: *)
f = Factor[f /. {x[2*n] -> x} /. {x[_] -> 1}]
(* This sequence (with initial zeros dropped): *)
CoefficientList[Series[f, {x, 0, 35}], x] (* L. Edson Jeffery, Oct 15 2017 *)

a(n)=n*(n-1)*(n^2-n+1)/6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = n*(n-1)*(n^2-n+1)/6.
a(n) = ((n^5 - (n-1)^5) - (n^1 - (n-1)^1))/30 = (n^5 - (n-1)^5 - 1)/30. - Xavier Acloque, Jan 25 2003
The partial sums of the octahedral numbers: a(n+1) = Sum_{i=0..n} A005900(i). - Jonathan Vos Post, Mar 14 2006
G.f.: -x^2*(x+1)^2/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013 [Which is just rephrasing the partial sum formula with the Murthy formula in A005900. - R. J. Mathar, Jun 14 2014]
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Yosu Yurramendi, Sep 03 2013
From Daniel Poveda Parrilla, Sep 09 2017: (Start)
a(n) = A169938(n-1)/6.
a(n+1) = A288486(n)/24. (End)
Sum_{n>=2} 1/a(n) = 12 - 2 * sqrt(3) * tanh(sqrt(3)*Pi/2). - Amiram Eldar, Jun 28 2020
E.g.f.: exp(x)*x^2*(3 + 4*x + x^2)/6. - Stefano Spezia, Dec 12 2021

A019298 Number of balls in pyramid with base either a regular hexagon or a hexagon with alternate sides differing by 1 (balls in hexagonal pyramid of height n taken from hexagonal close-packing).

Original entry on oeis.org

0, 1, 4, 11, 23, 42, 69, 106, 154, 215, 290, 381, 489, 616, 763, 932, 1124, 1341, 1584, 1855, 2155, 2486, 2849, 3246, 3678, 4147, 4654, 5201, 5789, 6420, 7095, 7816, 8584, 9401, 10268, 11187, 12159, 13186

Offset: 0

Eric E Blom (eblom(AT)REM.re.uokhsc.edu)

Alternately add and subtract successively longer sets of integers: 0; 1 = 0+1; -4 = 1-2-3; 11 = -4+4+5+6; -23 = 11-7-8-9-10; 42 = -23+11+12+13+14+15; -69 = 42-16-17-18-19-20-21; ... then take absolute values. - Walter Carlini, Aug 28 2003
Number of 3 X 3 symmetric matrices with nonnegative integer entries, such that every row (and column) sum equals n-1.
Equals Sum_{0..n} of "three-quarter squares" sequence (A077043). - Philipp M. Buluschek (kitschen(AT)romandie.com), Aug 12 2007
a(n) is the sum of the n-th row in A220075, n > 0. - Reinhard Zumkeller, Dec 03 2012
Sum of all the smallest parts in the partitions of 3n into three parts (see example). - Wesley Ivan Hurt, Jan 23 2014
For n > 0, a(n) is the number of (nonnegative integer) magic labelings of the prism graph Y_3 with magic sum n - 1. - L. Edson Jeffery, Sep 09 2017
Or number of magic labelings of LOOP X C_3 with magic sum n - 1, where LOOP is the 1-vertex, 1-loop-edge graph, as Y_k = I X C_k and LOOP X C_k have the same numbers of magic labelings when k is odd. - David J. Seal, Sep 13 2017
a(n) is the number of triples of integers in [1,n]^3 such that each pair has sum larger than n. - Bob Zwetsloot, Jul 23 2020

			Add last column for a(n) (n > 0).
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    1           4           11          23          42      ..  a(n)

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_3(lambda).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.14(a), p. 452.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, January 1999, p. 13.
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782. MR0201332 (34 #1216).
R. J. Mathar, Illustrations of magic labelings (2025)
R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, Section 5, F_3(x).
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A053493, A077043 (first differences), A002717.
Cf. A061927, A244497, A292281, A244873, A289992 (# of magic labelings of prism graph Y_k = I X C_k, for k = 4,5,6,7,8, up to an offset).
Cf. A006325, A244879, A244880 (# of magic labelings of LOOP X C_k, for k = 4,6,8, up to an offset).

[Floor((n^2+1)*(2*n+3)/8): n in [0..80]]; // Vincenzo Librandi, Jul 28 2013

Maple
```
series(x*(x^2+x+1)/(x+1)/(x-1)^4,x,80);
```

Table[ Ceiling[3*n^2/4], {n, 0, 37}] // Accumulate (* Jean-François Alcover, Dec 20 2012, after Philipp M. Buluschek's comment *)
CoefficientList[Series[x (x^2 + x + 1) / ((x + 1) (x - 1)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 28 2013 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 4, 11, 23}, 38] (* L. Edson Jeffery, Sep 09 2017 *)

a(n)=(n^2+1)*(2*n+3)\8 \\ Charles R Greathouse IV, Apr 04 2013

def A019298(n): return n*(n*(2*n+3)+2)+3>>3 # Chai Wah Wu, Jun 07 2025

a(n) = floor((n^2+1)(2n+3)/8).
G.f.: x*(x^2+x+1)/((x+1)*(x-1)^4).
a(n) = floor((2n^3 + 3n^2 + 2n)/8); also nearest integer to ((n+1)^4 - n^4)/16.
a(n) = (4n^3 + 6n^2 + 4n+1 - (-1)^n)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Mar 06 2004
a(n) = Sum_{i=1..n} i^2 - floor(i^2/4) = Sum_{i=1..n} i * (2n - 2i + 1 - floor((n - i + 1)/2) ). - Wesley Ivan Hurt, Jan 23 2014
E.g.f.: (1/16)*(-exp(-x) + exp(x)*(1 + 14*x + 18*x^2 + 4*x^3)). - Stefano Spezia, Nov 29 2019
a(2*n) = (1/2)*( n*(n + 1)^3 - (n - 1)*n^3 ); a(2*n-1) = (1/2)*( (n + 1)*n^3 - n*(n - 1)^3 ) (note: replacing the exponent 3 with 2 throughout gives the sequence of generalized pentagonal numbers A001318). - Peter Bala, Aug 11 2021
a(2n-1) = A213772(n). - R. J. Mathar, Mar 02 2025
(n-2)*a(n) -3*a(n-1) -(n+1)*a(n-2) +2*n-1 =0. - R. J. Mathar, Mar 09 2025

Error in n=8 term corrected May 15 1997

A050446 Table read by ascending antidiagonals: T(n, m) giving total degree of n-th-order elementary symmetric polynomials in m variables.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 4, 1, 1, 8, 14, 10, 5, 1, 1, 13, 31, 30, 15, 6, 1, 1, 21, 70, 85, 55, 21, 7, 1, 1, 34, 157, 246, 190, 91, 28, 8, 1, 1, 55, 353, 707, 671, 371, 140, 36, 9, 1, 1, 89, 793, 2037, 2353, 1547, 658, 204, 45, 10, 1, 1, 144, 1782, 5864, 8272, 6405, 3164, 1086, 285, 55, 11, 1

Offset: 0

N. J. A. Sloane, Dec 23 1999

T(n, m) is a polynomial of degree n in m. For example, T(2, m) = (m + 1)(m + 2)/2. For the polynomials corresponding to n = 1, 2, ..., 10, see the Cyvin-Gutman reference (p. 143). Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005
Let LOOP X C_k, k >= 1, be the graph constructed by attaching a loop to each vertex of the cycle graph C_k. Let G_n, n >= 0, be the graph obtained by deleting one edge from LOOP X C_{n+1} while retaining the n + 1 loops; e.g., for n = 4, see the graph G_4 at the top of the page in the Stanley link below. Then T(n, m) equals the number of magic labelings of G_n having magic sum m. (See the second Mathematica program below which requires the "Omega" package authored by Axel Riese and which can be downloaded from the link provided in the article by Andrews et al.) - L. Edson Jeffery, Oct 19 2017
For n != 1, T(n, m) is the number of up-down words of length n over an alphabet of size m. - Sela Fried, Apr 08 2025
Conjecture: T(n,m) is the number of words of length n over the alphabet [m] such that any pair of adjacent letters sum to at most m + 1. - John Tyler Rascoe, Jun 06 2025

			Array begins:
  [0]  1  1    1     1      1      1       1       1        1        1
  [1]  1  2    3     4      5      6       7       8        9       10
  [2]  1  3    6    10     15     21      28      36       45       55
  [3]  1  5   14    30     55     91     140     204      285      385
  [4]  1  8   31    85    190    371     658    1086     1695     2530
  [5]  1 13   70   246    671   1547    3164    5916    10317    17017
  [6]  1 21  157   707   2353   6405   15106   31998    62349   113641
  [7]  1 34  353  2037   8272  26585   72302  173502   377739   760804
  [8]  1 55  793  5864  29056 110254  345775  940005  2286648  5089282
  [9]  1 89 1782 16886 102091 457379 1654092 5094220 13846117 34053437
  ...
Triangle starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  2,  1;
  [3] 1,  3,  3,  1;
  [4] 1,  5,  6,  4,  1;
  [5] 1,  8, 14, 10,  5, 1;
  [6] 1, 13, 31, 30, 15, 6, 1;

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 142-144).

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Byeong-Gil Choe and Hyeong-Kwan Ju, A Recurrence Relation Associated with Unit-Primitive Matrices, arXiv:2305.03930 [math.CO], 2023.
L. Carlitz and R. Scoville, Up-down sequences, Duke Math. J. (39) (1972), 583-598.
Jane Ivy Coons and Seth Sullivant, The h*-polynomial of the order polytope of the zig-zag poset, arXiv:1901.07443 [math.CO], 2019.
Sela Fried, A formula for the number of up-down words, arXiv:2503.02005 [math.CO], 2025.
Emma L. L. Gao, Sergey Kitaev, and Philip B. Zhang, Pattern-avoiding alternating words, arXiv:1505.04078 [math.CO], 2015.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
Hyeong-Kwan Ju, On the sequence generated by a certain type of matrices, Honam Math. J. 39, No. 4, 665-675 (2017).
Daeseok Lee and H.-K. Ju, An Extension of Hibi's palindromic theorem, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
T. Kyle Petersen and Yan Zhuang, Zig-zag Eulerian polynomials, arXiv:2403.07181 [math.CO], 2024.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission] See p. 31.
Guoce Xin and Yueming Zhong, Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials, arXiv:2201.02376 [math.CO], 2022.

Rows give A000012, A000027, A000217, A000330, A006322, ...
Columns give A000012, A000045, A000045, A006356, A006357, A006358, ...
Cf. A050447.

A050446 := proc(n,m)
    option remember;
    if m=0 then
        1;
    else
        procname(n,m-1)+add( procname(2*k,m-1) *procname(n-1-2*k,m), k=0..floor((n-1)/2) );
    end if;
end proc:
for d from 0 to 12 do
    for m from 0 to d do
        printf("%d,",A050446(d-m,m)) ;
    end do:
end do: # R. J. Mathar, Dec 14 2011
A050446 := := (n, m) -> evalf(abs(add(tan(2*j*Pi/(2*m + 1))^2*sec(2*j*Pi/(2*m + 1))^(n - 1), j = 1 .. m))/(2^(n - 1)*(2*m + 1))): # Sela Fried, Apr 28 2025

t[n_, m_?Positive] := t[n, m] = t[n, m-1] + Sum[t[2k, m-1]*t[n-1 - 2k, m], {k, 0, (n-1)/2}]; t[n_, 0] = 1; Flatten[Table[t[i-k , k-1], {i, 1, 12}, {k, 1, i}]] (* Jean-François Alcover, Jul 25 2011, after formula *)
<< Omega.m; nmax = 9; Do[cond[n_] = {}; If[n == 0, cond[n] = {a[1] == a[2]}, AppendTo[cond[n], {a[1] + a[2] == a[2 n + 2], a[2 n] + a[2 n + 1] == a[2 n + 2]}]; If[n > 1, Do[AppendTo[cond[n], a[2 j] + a[2 j + 1] + a[2 j + 2] == a[2 n + 2]], {j, n - 1}]]]; cond[n] = Flatten[cond[n]]; f[n_] = OEqSum[Product[x[i]^a[i], {i, 2 n + 2}], cond[n], u][[1]] /. x[2 n + 2] -> y /. x[] -> 1; Do[f[n] = OEqR[f[n], Subscript[u, j]], {j, Length[cond[n]]}], {n, 0, nmax}]; Grid[Table[CoefficientList[Series[f[n], {y, 0, nmax}], y], {n, 0, nmax}]] (* _L. Edson Jeffery, Oct 19 2017 *)

from functools import cache
@cache
def T(n, k):
    return T(n, k - 1) + sum(T(2 * j, k - 1) * T(n - 1 - 2 * j, k)
        for j in range(1 + (n - 1) // 2)) if k > 0 else 1
for n in range(6): print([T(n - k, k) for k in range(n + 1)])
# Peter Luschny, Jun 08 2024

T(n, m) = T(n, m - 1) + Sum_{k=0..(n-1)/2} T(2*k, m - 1)*T(n - 1 - 2*k, m).
From Sela Fried, Apr 08 2025: (Start)
T(n, m) = 1/(2^(n-1)*(2*m+1))*|Sum_{j = 1..m} tan^2(2*j*Pi/(2*m+1))*sec^(n+1)(2*j*Pi/(2*m+1)))|.
G.f. for words of odd length over an alphabet of size m: x*U_{m-1}(1-x^2/2)/V_{m-1}(1-x^2/2),
g.f. for words of even length over an alphabet of size m: 1/V_{m-1}(1-x^2/2),
where U_k(x) and V_k(x) are the Chebyshev polynomials of the second and third kind, respectively. (End)

More terms from Naohiro Nomoto, Jul 03 2001

A006322 4-dimensional analog of centered polygonal numbers.

Original entry on oeis.org

1, 8, 31, 85, 190, 371, 658, 1086, 1695, 2530, 3641, 5083, 6916, 9205, 12020, 15436, 19533, 24396, 30115, 36785, 44506, 53383, 63526, 75050, 88075, 102726, 119133, 137431, 157760, 180265, 205096, 232408, 262361, 295120, 330855, 369741, 411958, 457691, 507130

Offset: 1

Albert Rich (Albert_Rich(AT)msn.com)

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Partial sums give A006414. - L. Edson Jeffery, Dec 13 2011
Also the number of (w,x,y,z) with all terms in {1,...,n} and w<=x>=y<=z, see A211795. - Clark Kimberling, May 19 2012

			An illustration for a(5)=190: 5*(1+2+3+4+5)+4*(2+3+4+5)+3*(3+4+5)+2*(4+5)+1*(5) gives 75+56+36+18+5=190. - _J. M. Bergot_, Feb 13 2018

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/4).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Plateau Polycubes and Lateral Area, arXiv:1811.05707 [math.CO], 2018. See Column 2 Table 2 p. 9.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]. See p. 31.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000330, A006414, A050446, A050447.

List([1..40], n->5*Binomial(n+2,4) + Binomial(n+1,2)); # Muniru A Asiru, Feb 13 2018

[n*(n+1)*(5*n^2 +5*n +2)/24: n in [1..40]]; // G. C. Greubel, Sep 02 2019

a:=n->5*binomial(n+2,4) + binomial(n+1,2): seq(a(n), n=1..40); # Muniru A Asiru, Feb 13 2018

Table[5*Binomial[n+2, 4] + Binomial[n+1, 2], {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[(1+3x+x^2)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Jun 09 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,8,31,85,190},40] (* Harvey P. Dale, Sep 27 2016 *)

a(n)=n*(5*n^3+10*n^2+7*n+2)/24 \\ Charles R Greathouse IV, Dec 13 2011, corrected by Altug Alkan, Aug 15 2017

[n*(n+1)*(5*n^2 +5*n +2)/24 for n in (1..40)] # G. C. Greubel, Sep 02 2019

a(n) = 5*C(n+2,4) + C(n+1,2) = (C(5*n+4,4) - 1)/5^3 = n*(n+1)*(5*n^2 + 5*n + 2)/24.
a(n) = (((n+1)^5-n^5) - ((n+1)^3-n^3))/24. - Xavier Acloque, Jan 14 2003, corrected by Eric Rowland, Aug 15 2017
Partial sums of A004068. - Xavier Acloque, Jan 15 2003
G.f.: x*(1+3*x+x^2)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = Sum_{i=1..n} Sum_{j=1..n} i * min(i,j). - Enrique Pérez Herrero, Jan 30 2013
a(n) = A000537(n) - A000332(n+2). - J. M. Bergot, Jun 03 2017
Sum_{n>=1} 1/a(n) = 42 - 4*sqrt(15)*Pi*tanh(sqrt(3/5)*Pi/2). - Amiram Eldar, May 28 2022
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: exp(x)*x*(2 + x)*(12 + 30*x + 5*x^2)/24.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

A050447 Table T(n,m) giving total degree of n-th-order elementary symmetric polynomials in m variables, -1 <= n, 1 <= m, transposed and read by upward antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 5, 1, 1, 5, 10, 14, 8, 1, 1, 6, 15, 30, 31, 13, 1, 1, 7, 21, 55, 85, 70, 21, 1, 1, 8, 28, 91, 190, 246, 157, 34, 1, 1, 9, 36, 140, 371, 671, 707, 353, 55, 1, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 1, 11, 55, 285, 1086, 3164, 6405, 8272, 5864, 1782, 144, 1

Offset: 0

N. J. A. Sloane, Dec 23 1999

			Table begins
.    1   1   1    1     1      1       1       1        1         1
.    1   2   3    5     8     13      21      34       55        89
.    1   3   6   14    31     70     157     353      793      1782
.    1   4  10   30    85    246     707    2037     5864     16886
.    1   5  15   55   190    671    2353    8272    29056    102091
.    1   6  21   91   371   1547    6405   26585   110254    457379
.    1   7  28  140   658   3164   15106   72302   345775   1654092
.    1   8  36  204  1086   5916   31998  173502   940005   5094220
.    1   9  45  285  1695  10317   62349  377739  2286648  13846117
.    1  10  55  385  2530  17017  113641  760804  5089282  34053437

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.

T. D. Noe, Table of 100 antidiagonals
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission] See p. 31.

Rows give A000012, A000045, A006356, A006357, A006358, ...
Columns give A000012, A000027, A000217, A000330, A006322, ...
Cf. A001924, A050446.

nmax = 12; t[n_, m_?Positive] := t[n, m] = t[n, m-1] + Sum[t[2k, m-1]*t[n-1-2k, m], {k, 0, (n-1)/2}]; t[n_, 0]=1; Flatten[ Table[ t[k-1, n-k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 14 2011 *)
nmax = 10; f[0, x_] := 1; f[1, x_] := 1/(1 - x); f[n_, x_] := (x + f[n - 2, x])/(1 - x^2 - x*f[n - 2, x]); t[n_, m_] := Coefficient[Series[f[n, x], {x, 0, m}], x, m]; Grid[Table[t[n, m], {n, nmax}, {m, 0, nmax - 1}]] (* L. Edson Jeffery, Oct 19 2017 *)

M(n)=matrix(n,n,i,j,if(sign(i+j-n)-1,0,1)); V(n)=vector(n,i,1); P(r,n)=vecmax(V(r)*M(r)^n) \\ P(r,n) is T(n,k); Benoit Cloitre, Jan 27 2003

See PARI code. See A050446 for recurrence.

G.f. for row n >= 0: f(n, x) = (x + f(n-2, x))/(1 - x^2 - x*f(n-2, x)), where f(0, x) = 1 and f(1, x) = 1/(1 - x) [R. P. Stanley]. - L. Edson Jeffery, Oct 19 2017

More terms from Naohiro Nomoto, Jul 03 2001

A205497 Triangle read by rows: Zig-zag Eulerian number triangle T(n, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 14, 31, 14, 1, 1, 26, 109, 109, 26, 1, 1, 46, 334, 623, 334, 46, 1, 1, 79, 937, 2951, 2951, 937, 79, 1, 1, 133, 2475, 12331, 20641, 12331, 2475, 133, 1, 1, 221, 6267, 47191, 123216, 123216, 47191, 6267, 221, 1

Offset: 0

L. Edson Jeffery, Jan 27 2012

From Kyle Petersen, Jun 02 2024: (Start)
Coefficients of the "P-Eulerian" polynomial of a naturally labeled zig-zag poset, which counts linear extensions according to number of descents. T(n, k) is the number of linear extensions of the n-element zig-zag poset with k descents.
Also T(n, k) is the number of up-down permutations of length n with k "big returns". A big return is a pair (i, i+1) for which i appears more than one place to the right of i+1 in the permutation. This interpretation implies row sums are given by A000111. (End)
From L. Edson Jeffery, Jan 27 2012: (Start)
(Previous name:) Omitting the first two ones, a rectangular array M read by antidiagonals in which entry M_{n-k, k} in row n-k and column k, 0 <= k <= n, gives the coefficient of x^k in the numerator of the conjectured generating function for row n + 3 of the tabular form of A050446.
In the following, let M_{n, k} denote the entry in row n and column k of M, n, k in {0, 1, ...}.
Conjecture: 1. M_{n, k} = M_{k, n}, for all n and k; that is, M is symmetric about the central terms {1, 3, 31, 623,...}. (This has been verified for the first 100 antidiagonals of M.)
Conjecture: 2. For m in {3, 4,...}, row m of array A050446 has generating function of the form H_m(x)/(1 - x)^m, in which the numerator H_m(x) is a polynomial of degree m - 3 in x with coefficients given by the entries of the (m - 3)-th antidiagonal of M containing the sequence of entries {M_{m-3-j,j}}, j=0..m-3 (see the example below). It is known that H_1(x) = H_2(x) = 1.
Conjecture: 3. Define the Chebyshev polynomials of the second kind by U_0(t) = 1, U_1(t) = 2*t and U_r(t) = 2*t*U_(r-1)(t) - U_(r-2)(t) (r > 1). Assuming Conjecture 1, lim_{n -> infinity} M_{n+1, k}/M_{n, k} = U_k(cos(Pi/(2*k+3))) = spectral radius of the (k+1) X (k+1) unit-primitive matrix (see [Jeffery]) A_{2*k+3, k} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], with identical limits for the columns of the transpose M^T of M.
Conjecture: 4. Let S(u, v) denote the entry in row u and column v of triangle S = A187660, 0 <= v <= u. Define the polynomials P_u(x) = Sum[S(u, v)*x^v]. Assuming Conjecture 1, then (i) the generating function for row (or column) n of M is of the form
G_n(x)/((P_1(x))^(n+1) * (P_2(x))^n * ... * (P_n(x))^2 * P_(n+1)(x)),
in which (ii) the numerator G_n(x) is a polynomial of degree A005586(n), and (iii) the denominator is a polynomial of degree A000292(n+1).
Remarks: The coefficients in the numerators G_n(x) appear to have no pattern. The polynomial P_j(x), j in {1,...,n+1}, of Conjecture 4 is also obtained from the characteristic polynomial of the unit-primitive matrix A_{2*j+3,j} of Conjecture 3 by taking the exponents of the latter in reverse order; and P_j(x) is otherwise identical to the characteristic polynomial of the unit-primitive matrix A_{2*j+3,1}.
(End)
Conjecture: The Eulerian zig-zag polynomials have only negative and simple real roots and form a Sturm sequence, that is, p(n+1, x) has n real roots separated by the roots of p(n, x). This property was proved by Frobenius for the classical Eulerian polynomials. - Peter Luschny, Jun 04 2024

			From _Kyle Petersen_, Jun 02 2024: (Start)
Triangle T(n, k) begins:
  1;
  1;
  1;
  1,  1;
  1,  3,   1;
  1,  7,   7,    1;
  1, 14,  31,   14,    1;
  1, 26, 109,  109,   26,   1;
  1, 46, 334,  623,  334,  46,  1;
  1, 79, 937, 2951, 2951, 937, 79, 1;
  ...
For n=4, the naturally labeled zig-zag poset 1<3>2<4 has five linear extensions: 1234, 1243, 2134, 2143, 2413, and their descent numbers are (respectively) 0, 1, 1, 2, 1. Thus T(4,0) = 1, T(4,1) = 3, and T(4,2) = 1. Also with n=4, there are five up-down permutations: 1324, 1423, 2314, 2413, 3412, and their big return numbers are (respectively) 0, 1, 1, 2, 1. (End)
Without the first two ones the data can be seen as an array M read by antidiagonals. Christopher H. Gribble kindly calculated the first 100 antidiagonals which starts as:
  1,  1,   1,     1,      1,       1, ...
  1,  3,   7,    14,     26,      46, ...
  1,  7,  31,   109,    334,     937, ...
  1, 14, 109,   623,   2951,   12331, ...
  1, 26, 334,  2951,  20641,  123216, ...
  1, 46, 937, 12331, 123216, 1019051, ...
  ...
The antidiagonals of M written as the rows of a triangle, yielding then, by the conjectures and the definition of H_m(x), row m = 7 of table A050446 has generating function H_7(x)/(1-x)^7 = (Sum_{j=0..4} M_{4-j,j}*x^j)/(1-x)^7 = (1 + 14*x + 31*x^2 + 14*x^3 + x^4)/(1-x)^7.

Jane Ivy Coons and Seth Sullivant, The h*-polynomial of the order polytope of the zig-zag poset, arXiv:1901.07443 [math.CO], 2019.
L. Edson Jeffery, Unit-primitive matrices
Hyeong-Kwan Ju, On the sequence generated by a certain type of matrices, Honam Math. J. 39, No. 4, 665-675 (2017).
Daeseok Lee and Hyeong-Kwan Ju, An Extension of Hibi's palindromic theorem, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
Peter Luschny, Illustrating the polynomials.
T. Kyle Petersen and Yan Zhuang, Zig-zag Eulerian polynomials, arXiv:2403.07181 [math.CO], 2024.
Igor Pak, Boris Shapiro, Ilya Smirnov, and Ken-ichi Yoshida, Hilbert-Kunz multiplicity of quadrics via the Ehrhart theory, Stockholm Univ. (Sweden, 2025). See p. 5.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission] See p. 31.
Guoce Xin and Yueming Zhong, Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials, arXiv:2201.02376 [math.CO], 2022.

Cf. A000012, A000292, A001924, A050446, A050447, A187660, row sums are A000111.

Cf. A350354, A373388, A373389.

Gn := proc(n) local F;
    if n = 0 then p*q*x/(1 - q*x);
    elif n > 0 then
        F := Gn(n - 1);
        simplify(p/(p - q)*(subs({p = q, q = p}, F) - subs(p = q, F)));
    fi;
end:
Zn := proc(n) expand(simplify(subs({p = 1, q = 1}, Gn(n))*(1 - x)^(n + 1))) end:
seq( coeffs(Zn(n)), n=0..15);  # Kyle Petersen, Jun 02 2024
# Alternative:
A205497row := proc(n) local k, j; ifelse(n < 2, 1,
seq(add((-1)^j * binomial(n + 1, j) * A050446(n, k - j), j = 0..k), k = 0..n-2)) end:  # Peter Luschny, Jun 17 2024

Gn[n_] := Module[{F}, If[n == 0, p*q*x/(1-q*x), If[n > 0, F = Gn[n-1]; Simplify[p/(p-q)*(ReplaceAll[F, {p -> q, q -> p}] - ReplaceAll[F, p -> q])]]]];
Zn[n_] := Expand[Simplify[ReplaceAll[Gn[n], {p -> 1, q -> 1}]*(1-x)^(n+1)]];
Table[Rest@CoefficientList[Zn[n], x], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jun 04 2024, after Kyle Petersen *)

from functools import cache
from math import comb as binomial
@cache
def S(n, k):
    return (S(n, k - 1) + sum(S(2 * j, k - 1) * S(n - 1 - 2 * j, k)
            for j in range(1 + (n - 1) // 2)) if k > 0 else 1)
def A205497(dim):  # returns [row(0), ..., row(dim-1)]
    if dim < 4: return [[1]] * dim
    Y = [[0 for _ in range(n - 2)] for n in range(dim + 1)]
    for n in range(dim + 1):
        for k in range(n - 2):
            for j in range(k + 1):
                Y[n][k] += (-1)**j * binomial(n, j) * S(n - 1, k - j)
    Y[1] = Y[2] = [1]
    return Y[1::]
print(A205497(9))  # Peter Luschny, Jun 14 2024

Conjecture: 5.1. G.f. for column 0 of M is 1/(1-x) (A000012).
Conjecture: 5.2. G.f. for column 1 of M is 1/((1-x)^2*(1-x-x^2)) (A001924).
Conjecture: 5.3. G.f. for column 2 of M is (1 - x^2 - x^3 - x^4 + x^5)/((1-x)^3*(1-x-x^2)^2*(1 - 2*x - x^2 + x^3)) (A205492).
Conjecture: 5.4. G.f. for column 3 of M is (1 + x - 6*x^2 - 15*x^3 + 21*x^4 + 35*x^5 - 13*x^6 - 51*x^7 + 3*x^8 + 21*x^9 + 5*x^10 + x^11 - 5*x^12 - x^13 - x^14)/((1-x)^4*(1-x-x^2)^3*(1 - 2*x - x^2 + x^3)^2*(1 - 2*x - 3*x^2 + x^3 + x^4)) (A205493).
Conjecture: 5.5. G.f. for column 4 of M is (1 + 4*x - 31*x^2 - 67*x^3 + 348*x^4 + 418*x^5 - 1893*x^6 - 1084*x^7 + 4326*x^8 + 4295*x^9 - 7680*x^10 - 9172*x^11 + 9104*x^12 + 11627*x^13 - 5483*x^14 - 10773*x^15 + 1108*x^16 + 7255*x^17 + 315*x^18 - 3085*x^19 - 228*x^20 + 669*x^21 + 102*x^22 - 23*x^23 - 45*x^24 - 16*x^25 + 11*x^26 + 2*x^27 - x^28)/((1-x)^5*(1-x-x^2)^4*(1 - 2*x - x^2 + x^3)^3*(1 - 2*x - 3*x^2 + x^3 + x^4)^2*(1 - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5)) (A205494).

Two 1's prepended and new name by Kyle Petersen Jun 02 2024

Edited by Peter Luschny, Jun 02 2024

A001496 Number of 4 X 4 matrices with nonnegative integer entries and row and column sums equal to n.

Original entry on oeis.org

1, 24, 282, 2008, 10147, 40176, 132724, 381424, 981541, 2309384, 5045326, 10356424, 20158151, 37478624, 66952936, 115479776, 193077449, 313981688, 498033282, 772409528, 1173759851, 1750812624, 2567527260, 3706873040

Offset: 0

N. J. A. Sloane

Number of 4 X 4 stochastic matrices of integers.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(4,r).
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).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, pages 233-234.
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.

T. D. Noe, Table of n, a(n) for n = 0..1000
A. G. Bell, Partitioning integers in n dimensions, The Computer Journal, 13 (1970), 278-283.
Brian Conrey and Alex Gamburd, Pseudomoments of the Riemann zeta-function and pseudomagic squares, Journal of Number Theory, Volume 117, Issue 2, April 2006, Pages 263-278.
I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
I. J. Good, On the application of symmetric Dirichlet distributions and contingency tables, pp. 1178-1179. (Annotated scanned copy)
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
J. N. van Rijn, F. W. Takes, J. K. Vis, Computing and Predicting Winning Hands in the Trick-Taking Game of Klaverjas, 30th Benelux Conference on Artificial Intelligence (BNAIC 2018), 's-Hertogenbosch, the Netherlands.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A002817, A003438, A019298.
See A002721 for a 3-dimensional analog.
Row n=4 of A257493.

CoefficientList[Series[(1 + 14*x + 87*x^2 + 148*x^3 + 87*x^4 + 14*x^5 + x^6)/(1 - x)^10, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 24 2017 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,24,282,2008,10147,40176,132724,381424,981541,2309384},30] (* Harvey P. Dale, Jul 12 2017 *)

x='x+O('x^99); Vec((1+14*x+87*x^2+148*x^3+87*x^4+14*x^5+x^6)/(1-x)^10) \\ Altug Alkan, Apr 17 2016

G.f.: (1+14*x+87*x^2+148*x^3+87*x^4+14*x^5+x^6)/(1-x)^10.

a(n) = binomial(n + 3, 3) + 20*binomial(n + 4, 5) + 152*binomial(n + 5, 7) + 352*binomial(n + 6, 9). [Equivalent to a formula given by Bell].

More terms from Vladeta Jovovic, Feb 06 2000

A244497 Number of magic labelings of the prism graph I X C_5 with magic sum n.

Original entry on oeis.org

1, 11, 57, 197, 533, 1223, 2494, 4654, 8105, 13355, 21031, 31891, 46837, 66927, 93388, 127628, 171249, 226059, 294085, 377585, 479061, 601271, 747242, 920282, 1123993, 1362283, 1639379, 1959839, 2328565, 2750815, 3232216, 3778776, 4396897, 5093387, 5875473, 6750813, 7727509, 8814119

Offset: 0

N. J. A. Sloane, Jul 07 2014

This sequence is also the number of magic labelings of the cycle-of-loops graph LOOP X C_5 with magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. A similar identity holds between the sequences for I X C_k and LOOP X C_k for all odd k. - David J. Seal, Sep 14 2017

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Cf. A019298, A061927, A292281, A244873, A289992 (analogs for prism graphs I X C_k, k = 3,4,6,7,8).

Cf. A006325, A244879, A244880 (analogs for LOOP X C_k, k = 4,6,8).

A244497:=n->(3*(63+(-1)^n) + 576*n + 720*n^2 + 460*n^3 + 150*n^4 + 20*n^5) / 192: seq(A244497(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

CoefficientList[Series[(1 + 6 x + 11 x^2 + 6 x^3 + x^4)/((1 - x)^6*(1 + x)), {x, 0, 37}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,11,57,197,533,1223,2494},40] (* Harvey P. Dale, Aug 04 2021 *)

Vec((1+6*x+11*x^2+6*x^3+x^4) / ((1-x)^6*(1+x)) + O(x^40)) \\ Colin Barker, Jan 13 2017

G.f.: (1 + 6*x + 11*x^2 + 6*x^3 + x^4) / ((1 - x)^6*(1 + x)).
From Colin Barker, Jan 13 2017: (Start)
a(n) = (3*(63+(-1)^n) + 576*n + 720*n^2 + 460*n^3 + 150*n^4 + 20*n^5) / 192.
a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7) for n>6.
(End)

A244873 Number of magic labelings of the prism graph I X C_7 with magic sum n.

Original entry on oeis.org

1, 29, 289, 1640, 6604, 21122, 57271, 137155, 298184, 599954, 1132942, 2029229, 3475465, 5728289, 9132418, 14141618, 21342771, 31483251, 45501823, 64563278, 90097018, 123839804, 167882881, 224723693, 297322402, 389163424, 504322196, 647537387, 824288767, 1040880947, 1304533204

Offset: 0

N. J. A. Sloane, Jul 08 2014

The graph is the 5th one shown in the link. This sequence is also the number of magic labelings of the cycle-of-loops graph LOOP X C_7 with magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. A similar identity holds between the sequences for I X C_k and LOOP X C_k for all odd k. - David J. Seal, Sep 14 2017

N. J. A. Sloane, Graphs for A244869-A244876.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (7, -20, 28, -14, -14, 28, -20, 7, -1).

Cf. A244869-A244876.
Cf. A019298, A061927, A244497, A292281, A289992 (analogs for prism graphs I X C_k, k = 3,4,5,6,8).
Cf. A006325, A244879, A244880 (analogs for LOOP X C_k, k = 4,6,8).

Table[61 n^7/1440 + 427 n^6/960 + 1463 n^5/720 + 2009 n^4/384 + 11809 n^3/1440 + 1253 n^2/160 + 169 n/40 + (-1)^n/256 + 255/256, {n, 0, 30}] (* Bruno Berselli, Jul 08 2014 *)
LinearRecurrence[{7,-20,28,-14,-14,28,-20,7,-1},{1,29,289,1640,6604,21122,57271,137155,298184},40] (* Harvey P. Dale, Aug 09 2017 *)

G.f.: (1+22*x+106*x^2+169*x^3+106*x^4+22*x^5+x^6)/((1-x)^8*(1+x)).

a(n) = 61*n^7/1440 + 427*n^6/960 + 1463*n^5/720 + 2009*n^4/384 + 11809*n^3/1440 + 1253*n^2/160 + 169*n/40 + (-1)^n/256 + 255/256. [Bruno Berselli, Jul 08 2014]

Name made more self-contained by David J. Seal, Sep 14 2017

cp's OEIS Frontend

A002817 Doubly triangular numbers: a(n) = n*(n+1)*(n^2+n+2)/8.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A006325 4-dimensional analog of centered polygonal numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A019298 Number of balls in pyramid with base either a regular hexagon or a hexagon with alternate sides differing by 1 (balls in hexagonal pyramid of height n taken from hexagonal close-packing).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A050446 Table read by ascending antidiagonals: T(n, m) giving total degree of n-th-order elementary symmetric polynomials in m variables.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A006322 4-dimensional analog of centered polygonal numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

A002817 Doubly triangular numbers: a(n) = n(n+1)(n^2+n+2)/8.