A106607 -id:A106607 - cp's OEIS frontend

A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.

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, 33, 29, 11, 1, 1, 1, 1, 4, 20, 74, 142, 79, 15, 1, 1, 1, 1, 5, 28, 163, 556, 742, 225, 22, 1, 1, 1, 1, 5, 39, 319, 1919, 5369, 4454, 677, 30, 1, 1

Offset: 0

Andrew Howroyd, Apr 08 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 A188403. Burnside's lemma as applied in A318805 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     20      28 ...
  4 | 1 1  5  12   33    74    163     319 ...
  5 | 1 1  7  29  142   556   1919    5793 ...
  6 | 1 1 11  79  742  5369  31781  156191 ...
  7 | 1 1 15 225 4454 64000 692599 5882230 ...
  ...
The T(3,3) = 5 matrices are:
   [0 0 3]  [0 1 2]  [0 1 2]  [1 0 2]  [1 1 1]
   [0 3 0]  [1 1 1]  [1 2 0]  [0 3 0]  [1 1 1]
   [3 0 0]  [2 1 0]  [2 0 1]  [2 0 1]  [1 1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..377
Ming Yean Lim, The number of irreducibles in the plethysm s_lambda[s_m], arXiv:2503.21108 [math.CO], 2025. See p. 8.

Rows n=0..5 are A000012, A000012, A008619, A106607, A333886, A333887.
Columns n=0..5 are A000012, A000012, A000041, A333888, A333889, A333890.
Main diagonal is A333738.
Cf. A188403 (labeled case), A333159 (binary), A333733 (not necessarily symmetric).
Cf. A318805, A333893.

A100779 Expansion of (1+t^2+4t^3+2t^4+t^5+3t^6)/((1-t)^2(1-t^2)*(1-t^3)^2).

Original entry on oeis.org

1, 2, 5, 14, 27, 49, 89, 142, 218, 329, 469, 651, 892, 1183, 1542, 1989, 2514, 3138, 3886, 4745, 5741, 6902, 8214, 9706, 11411, 13312, 15443, 17840, 20485, 23415, 26671, 30232, 34140, 38439, 43107, 48189, 53734, 59717, 66188, 73199, 80724, 88816, 97532, 106843

Offset: 0

N. J. A. Sloane, May 12 2005

Molien series for 5-dimensional group of order 48.

S. Ling and P. Solé, Type II Codes over F_4 + u F_4, European J. Combinatorics, 22 (2001), 983-997.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-3,0,3,0,0,-2,1).

Cf. A106607.

(1+t^2+4*t^3+2*t^4+t^5+3*t^6)/((1-t)^2*(1-t^2)*(1-t^3)^2);
seq(coeff(series(%,t,n+1),t,n), n=0..50);

A355759 Sums of the first ceiling((n+1)/2) entries on the diagonals of a square spiral with a starting value of 1 in the center, where the diagonal and the antidiagonal are used alternately.

Original entry on oeis.org

1, 4, 6, 11, 15, 24, 32, 45, 57, 76, 94, 119, 143, 176, 208, 249, 289, 340, 390, 451, 511, 584, 656, 741, 825, 924, 1022, 1135, 1247, 1376, 1504, 1649, 1793, 1956, 2118, 2299, 2479, 2680, 2880, 3101, 3321, 3564, 3806, 4071, 4335, 4624, 4912, 5225, 5537, 5876, 6214, 6579, 6943

Offset: 1

Karl-Heinz Hofmann, Aug 14 2022

			See the PDF in links.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, Examples of a(1..18)
Project Euler, Problem 28. Number spiral diagonals
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Cf. A200975, A106607, A114254.

Cf. A006527 and A208995 (bisections, see formulas).

CoefficientList[Series[-(x^7 - 2*x^6 + x^4 - x^3 + 2*x^2 - 2*x - 1)/((x - 1)^4*(x + 1)^2*(x^2 + 1)), {x, 0, 50}], x] (* Amiram Eldar, Aug 19 2022 *)

a(n) = (n^2 + 6*n + if(n%2,17,20))*n \ 24 + (n%4!=1); \\ Kevin Ryde, Aug 19 2022

def A355759(n):  # polynomial way.
    if   n % 2 == 0: return((24 + 20*n + 6*n**2 + n**3)//24)
    elif n % 4 == 3: return((12 + 17*n + 6*n**2 + n**3)//24)
    elif n % 4 == 1: return((     17*n + 6*n**2 + n**3)//24)

G.f.: -(x^7 - 2*x^6 + x^4 - x^3 + 2*x^2 - 2*x-1)/((x - 1)^4 * (x + 1)^2 * (x^2 +1)).
a(n) = (24 + 20*n + 6*n^2 + n^3) / 24  for n even.
a(n) = (12 + 17*n + 6*n^2 + n^3) / 24  for n odd and n (mod 4) == 3.
a(n) =      (17*n + 6*n^2 + n^3) / 24  for n odd and n (mod 4) == 1.
a(2*n) = A006527(n+1).
a(2*n-1) = A208995(n) - 1.
E.g.f.: ((30 + 45*x + 12*x^2 + x^3)*cosh(x) + (51 + 42*x + 12*x^2 + x^3)*sinh(x) - 6*cos(x))/24. - Stefano Spezia, Aug 19 2022

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A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.

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A100779 Expansion of (1+t^2+4t^3+2t^4+t^5+3t^6)/((1-t)^2(1-t^2)*(1-t^3)^2).

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A355759 Sums of the first ceiling((n+1)/2) entries on the diagonals of a square spiral with a starting value of 1 in the center, where the diagonal and the antidiagonal are used alternately.

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cp's OEIS Frontend

A333737 Array read by antidiagonals: T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k up to permutations of rows and columns.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A100779 Expansion of (1+t^2+4*t^3+2*t^4+t^5+3*t^6)/((1-t)^2*(1-t^2)*(1-t^3)^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A355759 Sums of the first ceiling((n+1)/2) entries on the diagonals of a square spiral with a starting value of 1 in the center, where the diagonal and the antidiagonal are used alternately.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A100779 Expansion of (1+t^2+4t^3+2t^4+t^5+3t^6)/((1-t)^2(1-t^2)*(1-t^3)^2).