A257493 -id:A257493 - 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

A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 24, 90, 24, 1, 1, 120, 2040, 2040, 120, 1, 1, 720, 67950, 297200, 67950, 720, 1, 1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1, 1, 40320, 187530840, 24046189440, 116963796250, 24046189440, 187530840, 40320, 1, 1, 362880, 14398171200, 12025780892160, 315031400802720, 315031400802720, 12025780892160, 14398171200, 362880, 1

Offset: 0

N. J. A. Sloane

Or, triangle of multipermutation numbers T(n,k), n >= 0, 0 <= k <= n: number of relations on an n-set such that all vertical sections and all horizontal sections have k elements.

			Triangle begins:
  1;
  1,    1;
  1,    2,       1;
  1,    6,       6,        1;
  1,   24,      90,       24,        1;
  1,  120,    2040,     2040,      120,       1;
  1,  720,   67950,   297200,    67950,     720,    1;
  1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,k).

Brendan D. McKay, Rows n = 0..30, flattened
C. J. Everett and P. R. Stein, The asymptotic number of integer stochastic matrices, Disc. Math. 1 (1971), 55-72.
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
Richard J. Mathar, Rencontres for equipartite distributions of multisets of colored balls into urns, vixra:2306.0157 (2023)
B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
Brendan D. McKay, first 30 rows : entries named Bv[n,k,n,k]
Wouter Meeussen, relevant entries from B. D. McKay reference

Row sums give A067209.
Central coefficients are A058527.
Cf. A000142 (column 1), A001499 (column 2), A001501 (column 3), A058528 (column 4), A075754 (column 5), A172544 (column 6), A172541 (column 7), A172536 (column 8), A172540 (column 9), A172535 (column 11), A172534 (column 12), A172538 (column 13), A172537 (column 14).
Cf. A133687, A333157 (symmetric matrices), A257493 (nonnegative elements), A260340 (up to row permutations), A364068 (traceless).

T(n, k)={
  local(M=Map(Mat([n, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(i, p, v, e) = if(i<0, if(!e, acc(p, v)), my(t=polcoef(p,i)); for(j=0, min(t, e), self()(i-1, p+j*(x-1)*x^i, binomial(t, j)*v, e-j))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[,2]);
} \\ Andrew Howroyd, Apr 03 2020

Comtet quotes Everett and Stein as showing that T(n,k) ~ (kn)!(k!)^(-2n) exp( -(k-1)^2/2 ) for fixed k as n -> oo.

T(n,k) = T(n,n-k).

More terms from Greg Kuperberg, Feb 08 2001

A321719 Number of non-normal semi-magic squares with sum of entries equal to n.

Original entry on oeis.org

1, 1, 3, 7, 28, 121, 746, 5041, 40608, 362936, 3635017, 39916801, 479206146, 6227020801, 87187426839, 1307674521272, 20923334906117, 355687428096001, 6402415241245577, 121645100408832001, 2432905938909013343, 51090942176372298027, 1124001180562929946213

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Squares must be of size k X k where k is a divisor of n. This implies that a(p) = p! + 1 for p prime since the only allowable squares are of sizes 1 X 1 and p X p. The 1 X 1 square is [p], the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 1 for n > 1. - Chai Wah Wu, Jan 13 2019

			The a(3) = 7 semi-magic squares:
  [3]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Andrew Howroyd, Table of n, a(n) for n = 0..100
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A120732, A120733, A257493, A271103, A319056, A319616.

Cf. A321717, A321718, A321720, A321721, A321722, A321723, A321724.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(p) = p! + 1 for p prime and a(n) >= n! + 1 for n > 1 (see comment above). - Chai Wah Wu, Jan 13 2019

a(n) = Sum_{d|n} A257493(d, n/d) for n > 0. - Andrew Howroyd, Apr 11 2020

a(7) from Chai Wah Wu, Jan 13 2019
a(6) corrected and a(8)-a(15) added by Chai Wah Wu, Jan 14 2019
a(16)-a(19) from Chai Wah Wu, Jan 16 2019
Terms a(20) and beyond from Andrew Howroyd, Apr 11 2020

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

A202784 T(n,k) is the number of n X n 0..k arrays with row and column sums equal.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 14, 1, 1, 5, 16, 87, 140, 1, 1, 6, 25, 340, 5673, 4322, 1, 1, 7, 36, 1001, 89520, 2577513, 434542, 1, 1, 8, 49, 2442, 790425, 290838284, 8459809773, 144109562, 1, 1, 9, 64, 5215, 4756140, 11991502235, 12092687195284, 207495141083975, 165431317452, 1

Offset: 0

R. H. Hardin, Dec 24 2011

			Table starts
n\k | 0            1               2              3                4
----+-------------------------------------------------------------------
  0 | 1            1               1              1                1 ...
  1 | 1            2               3              4                5 ...
  2 | 1            4               9             16               25 ...
  3 | 1           14              87            340             1001 ...
  4 | 1          140            5673          89520           790425 ...
  5 | 1         4322         2577513      290838284      11991502235 ...
  6 | 1       434542      8459809773 12092687195284 3632894572698505 ...
  7 | 1    144109562 207495141083975 ...
  8 | 1 165431317452 ...
  ...
Some solutions for n=4 k=3
..3..1..2..1....2..2..0..3....1..1..1..2....1..3..0..0....1..2..0..2
..2..3..2..0....1..2..2..2....3..0..1..1....1..0..1..2....3..1..0..1
..2..2..0..3....3..0..3..1....0..3..2..0....2..0..1..1....0..0..3..2
..0..1..3..3....1..3..2..1....1..1..1..2....0..1..2..1....1..2..2..0

Andrew Howroyd, Table of n, a(n) for n = 0..119 (first 15 antidiagonals; terms up to n=61 from R. H. Hardin)

Rows n=0..5 are A000012, A000027(k+1), A000290(k+1), A202785, A202786, A202787.
Columns k=1..15 are A067209, A067210, A067211, A067212, A067213, A067214, A067215, A067216, A067217, A067218, A067219, A067220, A067221, A067222, A067223.
Cf. A008300, A257493.

Added crossrefs. - R. H. Hardin, Jan 05 2012

Row and column zero inserted by Andrew Howroyd, Oct 14 2024

A000681 Number of n X n matrices with nonnegative entries and every row and column sum 2.

Original entry on oeis.org

1, 1, 3, 21, 282, 6210, 202410, 9135630, 545007960, 41514583320, 3930730108200, 452785322266200, 62347376347779600, 10112899541133589200, 1908371363842760216400, 414517594539154672566000, 102681435747106627787376000, 28772944645196614863048048000

Offset: 0

N. J. A. Sloane, Simon Plouffe

Or, number of labeled 2-regular pseudodigraphs (multiple arcs and loops allowed) of order n.

Also, number of permutations of the multiset {1^2,2^2,...,n^2} with the descent set consisting of multiples of 2. - Max Alekseyev, Apr 28 2014

			G.f. = 1 + x + 3*x^2 + 21*x^3 + 282*x^4 + 6210*x^5 + 202410*x^6 + 9135630*x^7 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 125, #25, a_n.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, section 3.5.10.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
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, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (a).
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.
C. B. Tompkins, Methods of successive restrictions in computational problems involving discrete variables. 1963, Proc. Sympos. Appl. Math., Vol. XV pp. 95-106; Amer. Math. Soc., Providence, R.I.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 49 terms from R. W. Robinson)
H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1996), 757-769.
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
S. Cockburn and J. Lesperance, Deranged socks, Mathematics Magazine, 86 (2013), 97-109.
Ira Gessel, Enumerative applications of symmetric functions, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.
William George Griffiths, On Integer Solutions to Linear Equations, Annals of Combinatorics 12:1 (2008), pp. 53-70.
Rui-Li Liu, Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
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]
Index entries for sequences related to magic squares

Column k=2 of A257493.
Row sums of A269742 and A307804.
Row and column sums equal s: A000142 (s=1), A001500 (s=3), A172806 (s=4), A172862 (s=5), A172894 (s=6), A172919 (s=7), A172944 (s=8), A172958 (s=9).
Cf. A001499, A005650, A123544.

A000681 := proc(n)
    coeftayl( exp(x/2)/sqrt(1-x),x=0,n) ;
    %*(n!)^2 ;
end proc:
seq(A000681(n),n=0..10) ; # R. J. Mathar, May 01 2019

a[n_] := Sum[ ((2*i)!*n!^2) / (2^i*(i!^2*(n - i)!)), {i, 0, n}]/2^n; Table[ a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 08 2011 *)
a[n_] := n!*HypergeometricPFQ[{1/2, -n}, {}, -2]/2^n; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 08 2012 *)

Vec( serlaplace(serlaplace( exp(x/2)/sqrt(1-x) )) ) /* Max Alekseyev, Apr 28 2014 */

from sage.combinat.integer_matrices import IntegerMatrices
def a(n): return IntegerMatrices([2]*n, [2]*n).cardinality() # Ralf Stephan, Mar 02 2014

Sum_{n >= 0} a(n) x^n / n!^2 = exp(x/2) / sqrt(1-x).
D-finite with recurrence a(n) = n^2*a(n-1) - (1/2)*n*(n-1)^2*a(n-2).
a(n) is asymptotic to c/sqrt(n)*(n!)^2 where c=0.93019... - Benoit Cloitre, Jun 25 2004
a(n) = sum(i=0..n, 2^(i-2*n) * C(n, i)^2 * (2*n-2*i)! * i! ).
a(n) = 2^(-n) * sum(i=0..n, ((n!)^2*(2*i)!) / ((i!)^2*((n-i)!*2^i)) ). - Shanzhen Gao, Nov 05 2007
In Cloitre's formula is c = exp(1/2)/sqrt(Pi) = 0.9301913671026328586. - Vaclav Kotesovec, Aug 12 2013
With c as used above by Cloitre and Kotesovec, a(n) is asymptotic to c/sqrt(n)*(n!)^2 * (1 + 2/(16*n) + 50/(16*n)^2 + 1100/(16*n)^3 + 32438/(16*n)^4 + 1185660/(16*n)^5 + 50498228/(16*n)^6 + 2438464600/(16*n)^7 + 131323987366/(16*n)^8 + 7782036656108/(16*n)^9 + 501905392385436/(16*n)^10 + ...). - Jon E. Schoenfield, Mar 03 2014
E.g.f.: 2/((2-x)*W(0)), where W(k) = 1 - (2*k+1)*x/(2-x-2*(k+1)*x/W(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2014

More terms from David W. Wilson

A188403 T(n,k) = Number of (n*k) X k binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 10, 11, 4, 1, 26, 56, 23, 5, 1, 76, 348, 214, 42, 6, 1, 232, 2578, 2698, 641, 69, 7, 1, 764, 22054, 44288, 14751, 1620, 106, 8, 1, 2620, 213798, 902962, 478711, 62781, 3616, 154, 9, 1, 9496, 2313638, 22262244, 20758650, 3710272, 222190, 7340, 215, 10, 1

Offset: 1

R. H. Hardin, Mar 30 2011

From Andrew Howroyd, Apr 09 2020: (Start)
T(n,k) is the number of k X k symmetric matrices with nonnegative integer entries and all row and column sums n. The number of such matrices up to isomorphism is given in A333737.
T(n,k) is also the number of loopless multigraphs with k labeled nodes of degree n or less. The number of such multigraphs up to isomorphism is given in A333893. (End)

			Table starts
  1  2   4    10      26        76         232          764          2620
  1  3  11    56     348      2578       22054       213798       2313638
  1  4  23   214    2698     44288      902962     22262244     648446612
  1  5  42   641   14751    478711    20758650   1158207312   80758709676
  1  6  69  1620   62781   3710272   313568636  36218801244 5518184697792
  1  7 106  3616  222190  22393101  3444274966 767013376954 ...
  1  8 154  7340  681460 111200600 29445929253 ...
  1  9 215 13825 1865715 472211360 ...
  1 10 290 24510 4655535 ...
  1 11 381 41336 ...
  ...
All solutions for 4 X 2:
..1..0....1..1....1..1
..1..0....1..1....1..0
..0..1....0..0....0..1
..0..1....0..0....0..0

Andrew Howroyd, Table of n, a(n) for n = 1..351 (first 95 terms from R. H. Hardin; terms 96..153 from Alois P. Heinz)
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014

Columns 1..8 are A000012, A000027(n+1), A019298(n+1), A053493, A053494, A188400, A188401, A188402.
Rows 1..8 are A000085, A000985, A188404, A188405, A188406, A188407, A188408, A188409.
Main diagonal is A333739.
Cf. A257493, A333157, A333737, A333893.

T(k,n)={
  local(M=Map(Mat([0, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(r, h, p, q, v, e) = if(!p, acc(x^e+q, v), my(i=poldegree(p), t=pollead(p)); self()(r, k, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, (k-e)\m), self()(r, if(j==t, k, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e+j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-r, k, src[i, 1], 0, src[i, 2], 0))); vecsum(Mat(M)[,2]);
}
{for(n=1, 7, for(k=1, 7, print1(T(n,k),", ")); print)} \\ Andrew Howroyd, Apr 08 2020

A333733 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer matrices with all row and column sums equal to k up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 5, 1, 1, 1, 1, 3, 9, 12, 7, 1, 1, 1, 1, 4, 13, 43, 31, 11, 1, 1, 1, 1, 4, 22, 106, 264, 103, 15, 1, 1, 1, 1, 5, 30, 321, 1856, 2804, 383, 22, 1, 1, 1, 1, 5, 45, 787, 12703, 65481, 44524, 1731, 30, 1, 1

Offset: 0

Andrew Howroyd, Apr 04 2020

Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A257493. Burnside's lemma can be used to extend this method to the unlabeled case.

			Array begins:
=======================================================
n\k | 0 1  2   3     4       5         6          7
----+--------------------------------------------------
  0 | 1 1  1   1     1       1         1          1 ...
  1 | 1 1  1   1     1       1         1          1 ...
  2 | 1 1  2   2     3       3         4          4 ...
  3 | 1 1  3   5     9      13        22         30 ...
  4 | 1 1  5  12    43     106       321        787 ...
  5 | 1 1  7  31   264    1856     12703      71457 ...
  6 | 1 1 11 103  2804   65481   1217727   16925049 ...
  7 | 1 1 15 383 44524 3925518 224549073 8597641912 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)

Rows n=0..5 are A000012, A000012, A008619, A052282, A052280, A333735.
Columns k=0..5 are A000012, A000012, A000041, A232215, A232216, A333736.
Main diagonal is A333734.
Cf. A133687, A167625, A257493, A333330, A377060.

A333901 Array read by antidiagonals: T(n,k) is the number of n X k nonnegative integer matrices with all column sums n and row sums k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 55, 19, 1, 1, 1, 1, 51, 415, 415, 51, 1, 1, 1, 1, 141, 3391, 10147, 3391, 141, 1, 1, 1, 1, 393, 28681, 261331, 261331, 28681, 393, 1, 1, 1, 1, 1107, 248137, 7100821, 22069251, 7100821, 248137, 1107, 1, 1

Offset: 0

Andrew Howroyd, Apr 18 2020

T(n,k) is the number of ordered factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity).

			Array begins:
=======================================================
n\k | 0 1   2     3       4          5            6
----+--------------------------------------------------
  0 | 1 1   1     1       1          1            1 ...
  1 | 1 1   1     1       1          1            1 ...
  2 | 1 1   3     7      19         51          141 ...
  3 | 1 1   7    55     415       3391        28681 ...
  4 | 1 1  19   415   10147     261331      7100821 ...
  5 | 1 1  51  3391  261331   22069251   1985311701 ...
  6 | 1 1 141 28681 7100821 1985311701 602351808741 ...
  ...
The T(3,2) = 7 matrices are:
  [1 1]  [1 1]  [1 1]  [2 0]  [2 0]  [0 2]  [0 2]
  [1 1]  [2 0]  [0 2]  [1 1]  [0 2]  [1 1]  [2 0]
  [1 1]  [0 2]  [2 0]  [0 2]  [1 1]  [2 0]  [1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..405 (antidiagonals n=0..27)

Columns k=0..9 are A000012, A000012, A002426, A172743, A172816, A172868, A172904, A172931, A172947, A172961.
Main diagonal is A110058.
Cf. A257462, A257493.

T(n, k)={
  local(M=Map(Mat([k, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(n, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, e\m), self()(if(j==t, n, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n, src[i, 1], 0, src[i, 2], k))); vecsum(Mat(M)[, 2])
}
for(n=0, 7, for(k=0, 7, print1(T(n,k), ", ")); print)

T(n,k) = T(k,n).

A001500 Number of stochastic matrices of integers: n X n arrays of nonnegative integers with all row and column sums equal to 3.

Original entry on oeis.org

1, 1, 4, 55, 2008, 153040, 20933840, 4662857360, 1579060246400, 772200774683520, 523853880779443200, 477360556805016931200, 569060910292172349004800, 868071731152923490921728000, 1663043727673392444887284377600, 3937477620391471128913917360384000

Offset: 0

N. J. A. Sloane

Also, number of bicubical multigraphs on 2n labeled nodes of two colors [Read, 1958, 1971]. - N. J. A. Sloane, Sep 09 2014

			a(2) = 4 with: [0 3]    [1 2]    [2 1]    [3 0]
               [3 0],   [2 1],   [1 2],   [0 3]. - _Bernard Schott_, Oct 15 2019

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 125, Problem 25(4), b_n (but beware errors).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
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).
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.

Alois P. Heinz, Table of n, a(n) for n = 0..180
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Petter Brändén, Jonathan Leake, Igor Pak, Lower bounds for contingency tables via Lorentzian polynomials, arXiv:2008.05907 [math.CO], 2020.
I. P. Goulden, D. M. Jackson, and J. W. Reilly, The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence, although there are errors)
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]
Index entries for sequences related to magic squares

Row sums of A269743 and of A344379.
Column k=3 of A257493.
Cf. A000681, A246970.

a[n_] := 6^(-n) Sum[2^j 3^k n!^2 (3n - 2k - 3j)!/(j! k! (n - j - k)!^2 * 6^(n - j - k)), {j, 0, n}, {k, 0, n - j}];
a /@ Range[0, 15] (* Jean-François Alcover, Oct 15 2019, after Shanzhen Gao *)

From Vladeta Jovovic, Mar 26 2001: (Start)
E.g.f. y(x) = Sum_{n >= 0} a(n)*x^n/(n!)^2 satisfies differential equation 81*x^5*(x^4 - x^2 + x + 4)*(d^4/dx^4)y(x) + 324*x^4*(x^4 - x^2 + x + 4)*(d^3/dx^3)y(x) - 9*x*(x^10 - 4*x^9 + 22*x^8 - 8*x^7 - 22*x^6 + 8*x^5 + 106*x^4 + 234*x^3 + 48*x^2 - 320*x + 64)*(d^2/dx^2)y(x) - 9*(x^10 - 4*x^9 + 22*x^8 - 8*x^7 - 4*x^6 + 8*x^5 + 88*x^4 + 252*x^3 + 120*x^2 - 320*x + 64)*(d/dx)y(x) + (x^11 - 7*x^10 + 30*x^9 - 16*x^8 - 43*x^7 + 51*x^6 + 238*x^5 + 630*x^4 + 36*x^3 - 1944*x^2 - 1152*x + 576)*y(x) = 0.
Recurrence: a(n) = n!*v(n) where v(n) = 1/(576*n)*((-198*n^9 + 8712*n^8 - 165175*n^7 + 1764196*n^6 - 11643772*n^5 + 48965728*n^4 - 130257475*n^3 + 209370724*n^2 - 182126340*n + 64083600)*v(n - 8) + (36*n^10 - 1944*n^9 + 45884*n^8 - 621504*n^7 + 5330892*n^6 - 30123576*n^5 + 112954596*n^4 - 275612976*n^3 + 415021552*n^2 - 343920960*n + 116928000)*v(n - 9) + (-9*n^11 + 585*n^10 - 16800*n^9 + 280800*n^8 - 3027357*n^7 + 22034565*n^6 - 110039130*n^5 + 375129450*n^4 - 849926784*n^3 + 1208298600*n^2 - 958439520*n + 315705600)*v(n - 10) + (-7*n^10 + 385*n^9 - 9240*n^8 + 127050*n^7 - 1104411*n^6 + 6314385*n^5 - 23918510*n^4 + 58866500*n^3 - 89275032*n^2 + 74400480*n - 25401600)*v(n - 11) + (-81*n^7 + 1944*n^6 - 20232*n^5 + 115578*n^4 - 383283*n^3 + 724230*n^2 - 708372*n + 270216)*v(n - 4) + (-72*n^6 + 1440*n^5 - 10890*n^4 + 40500*n^3 - 78678*n^2 + 75780*n - 28080)*v(n - 5) + (81*n^9 - 3321*n^8 + 59004*n^7 - 594054*n^6 + 3718687*n^5 - 14927199*n^4 + 38152096*n^3 - 59311746*n^2 + 50236612*n - 17330160)*v(n - 6) + (72*n^8 - 2520*n^7 + 37347*n^6 - 304479*n^5 + 1484133*n^4 - 4394565*n^3 + 7642248*n^2 - 7039116*n + 2576880)*v(n - 7) + (n^11 - 66*n^10 + 1925*n^9 - 32670*n^8 + 357423*n^7 - 2637558*n^6 + 13339535*n^5 - 45995730*n^4 + 105258076*n^3 - 150917976*n^2 + 120543840*n - 39916800)*v(n - 12) + (2880*n^2 - 5760*n + 3456)*v(n - 1) + (324*n^5 - 3564*n^4 + 14148*n^3 - 26028*n^2 + 21312*n - 6192)*v(n - 2) + (81*n^6 - 1377*n^5 + 7209*n^4 - 13203*n^3 - 3402*n^2 + 32076*n - 21384)*v(n - 3)). (End)
a(n) = 6^(-n) * Sum_{ alpha = 0..n, beta = 0..n-alpha } (2^alpha*3^beta*(n!)^2*(-2*beta+3*n-3*alpha)!)/(alpha!*beta!*(n-alpha-beta)!^2*6^(n-alpha-beta)). - Shanzhen Gao, Nov 05 2007
a(n) ~ sqrt(Pi) * 3^(n + 1/2) * n^(3*n + 1/2) / (2^(2*n - 1/2) * exp(3*n - 2)). - Vaclav Kotesovec, Oct 15 2019

More terms from Vladeta Jovovic, Mar 26 2001

cp's OEIS Frontend

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

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A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

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A321719 Number of non-normal semi-magic squares with sum of entries equal to n.

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

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A202784 T(n,k) is the number of n X n 0..k arrays with row and column sums equal.

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A000681 Number of n X n matrices with nonnegative entries and every row and column sum 2.

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