A053494 -id:A053494 - cp's OEIS frontend

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.

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

A053493 Number of symmetric 4 X 4 matrices of nonnegative integers with every row and column adding to n.

Original entry on oeis.org

1, 10, 56, 214, 641, 1620, 3616, 7340, 13825, 24510, 41336, 66850, 104321, 157864, 232576, 334680, 471681, 652530, 887800, 1189870, 1573121, 2054140, 2651936, 3388164, 4287361, 5377190, 6688696, 8256570, 10119425, 12320080, 14905856, 17928880, 21446401

Offset: 0

N. J. A. Sloane, Jan 15 2000; definition revised Jul 06 2014

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_4(lambda).

Colin Barker, Table of n, a(n) for n = 0..1000
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782. MR0201332 (34 #1216).
R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, F_4(x) in section 5.
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Cf. A019298, A053494.

CoefficientList[Series[(1+4x+10x^2+4x^3+x^4)/((1-x)^7(1+x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{1,10,56,214,641,1620,3616,7340},30] (* Harvey P. Dale, Oct 31 2011 *)

Vec((1+4*x+10*x^2+4*x^3+x^4) / ((1-x)^7*(1+x)) + O(x^40)) \\ Colin Barker, Jan 14 2017

G.f.: (1+4*x+10*x^2+4*x^3+x^4)/((1-x)^7*(1+x)).
a(0)=1, a(1)=10, a(2)=56, a(3)=214, a(4)=641, a(5)=1620, a(6)=3616, a(7)=7340, a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8). - Harvey P. Dale, Oct 31 2011
a(n) = (9*(31+(-1)^n) + 768*n + 928*n^2 + 624*n^3 + 238*n^4 + 48*n^5 + 4*n^6) / 288. - Colin Barker, Jan 14 2017

A070212 Number of 5 X 5 pandiagonal magic squares with sum n.

Original entry on oeis.org

1, 10, 55, 220, 715, 2001, 4995, 11385, 24090, 47905, 90376, 162955, 282490, 473110, 768570, 1215126, 1875015, 2830620, 4189405, 6089710, 8707501, 12264175, 17035525, 23361975, 31660200, 42436251, 56300310, 73983205, 96354820, 124444540, 159463876, 202831420, 256200285, 321488190

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 07 2002

In contrast to other definitions, a magic square may contain here any nonnegative integers, not necessarily distinct. For example, the 10 solutions for n = 1 are the 10 permutation matrices of size 5 X 5 which are pandiagonal in the sense that any of the 10 (principal or broken) diagonals has exactly one 1 and four 0's. - M. F. Hasler, Oct 23 2018

Muniru A Asiru, Table of n, a(n) for n = 0..5000
M. Ahmed, J. De Loera, and R. Hemmecke, Polyhedral Cones of Magic Cubes and Squares, arXiv:math/0201108 [math.CO], 2002.
Maya Ahmed, Jesús De Loera and Raymond Hemmecke, Polyhedral cones of magic cubes and squares, in Discrete and Computational Geometry, Springer, Berlin, 2003, pp. 25-41.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
Index entries related to magic squares.

Cf. A027567, A014820, A111158, A053494.

a:=[1, 10, 55, 220, 715, 2001, 4995, 11385, 24090];;  for n in [10..36] do a[n]:=9*a[n-1]-36*a[n-2]+84*a[n-3]-126*a[n-4]+126*a[n-5]-84*a[n-6]+36*a[n-7]-9*a[n-8]+a[n-9]; od; a; # Muniru A Asiru, Oct 23 2018

seq(coeff(series(-(x^4+x^3+x^2+x+1)/(x-1)^9,x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 23 2018

LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,10,55,220,715,2001,4995,11385,24090},40] (* Harvey P. Dale, Mar 13 2018 *)

apply( A070212(n)=1/8064*(n+4)*(n+3)*(n+2)*(n+1)*(n^2+5*n+8)*(n^2+5*n+42), [0..20]) \\ Edited by M. F. Hasler, Oct 23 2018

a(n) = (1/8064) * (n+4)*(n+3)*(n+2)*(n+1)*(n^2+5n+8)*(n^2+5n+42).

G.f.: -(x^4+x^3+x^2+x+1) / (x-1)^9. [Colin Barker, Dec 10 2012]

More terms from Benoit Cloitre, May 12 2002

More terms from M. F. Hasler, Oct 23 2018

A244868 Number of symmetric 5 X 5 matrices of nonnegative integers with zeros on the main diagonal and every row and column adding to n.

Original entry on oeis.org

1, 22, 158, 654, 1980, 4906, 10577, 20588, 37059, 62710, 100936, 155882, 232518, 336714, 475315, 656216, 888437, 1182198, 1548994, 2001670, 2554496, 3223242, 4025253, 4979524, 6106775, 7429526, 8972172, 10761058, 12824554, 15193130, 17899431, 20978352, 24467113, 28405334, 32835110, 37801086

Offset: 0

N. J. A. Sloane, Jul 07 2014

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 (6,-15,20,-15,6,-1).

Even bisection of row n=5 of A333351.

Cf. A053494.

Vec((1 + 16*x + 41*x^2 + 16*x^3 + x^4) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 16*x + 41*x^2 + 16*x^3 + x^4) / (1 - x)^6.
From Colin Barker, Jan 11 2017: (Start)
a(n) = (24 + 94*n + 165*n^2 + 155*n^3 + 75*n^4 + 15*n^5) / 24.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
(End)

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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.

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A053493 Number of symmetric 4 X 4 matrices of nonnegative integers with every row and column adding to n.

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A070212 Number of 5 X 5 pandiagonal magic squares with sum n.

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A244868 Number of symmetric 5 X 5 matrices of nonnegative integers with zeros on the main diagonal and every row and column adding to n.

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