A028657 -id:A028657 - cp's OEIS frontend

A052265 Triangle giving T(n,r) = number of equivalence classes of Boolean functions of n variables and range r=0..2^n under action of symmetric group.

1, 1, 1, 2, 1, 1, 3, 4, 3, 1, 1, 4, 9, 16, 20, 16, 9, 4, 1, 1, 5, 17, 52, 136, 284, 477, 655, 730, 655, 477, 284, 136, 52, 17, 5, 1, 1, 6, 28, 134, 625, 2674, 10195, 34230, 100577, 258092, 579208, 1140090, 1974438, 3016994, 4077077, 4881092, 5182326, 4881092

Offset: 0

Vladeta Jovovic, Feb 04 2000

Also, T(n,k) is the number of unlabeled n-vertex hypergraphs (or set systems) with k hyperedges. - Pontus von Brömssen, Apr 10 2024

			Triangle begins:
   1, 1;
   1, 2, 1;
   1, 3, 4, 3, 1;
   1, 4, 9, 16, 20, 16, 9, 4, 1;
   1, 5, 17, 52, 136, 284, 477, 655, 730, 655, 477, 284, 136, 52, 17, 5, 1;
   ...

M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 147.

Andrew Howroyd, Table of n, a(n) for n = 0..2057
Index entries for sequences related to Boolean functions

Row sums give A003180.

Cf. A028657, A371830 (empty hyperedge not permitted).

Table[rl = Table[Tuples[{0, 1}, nn][[i]] -> i, {i, 1, 2^nn}];
 f[permutation_] := PermutationCycles[Map[Permute[#, permutation] &, Tuples[{0, 1}, nn]] /. rl];CoefficientList[(Map[CycleIndexPolynomial[#, Array[Subscript[x, ##] &, 2^nn],2^nn] &, Map[f, Permutations[Range[nn]]]] // Total)/nn! /.
Table[Subscript[x, i] -> 1 + x^i, {i, 1, nn!}], x], {nn, 0, 8}] (* Geoffrey Critzer, Jun 22 2021 *)

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
Fix(q,x)={my(v=divisors(lcm(Vec(q))), u=apply(t->2^sum(j=1, #q, gcd(t, q[j])), v)); prod(i=1, #v, my(t=v[i]); (1+x^t)^(sum(j=1, i, my(d=t/v[j]); if(!frac(d), moebius(d)*u[j]))/t))}
Row(n)={my(s=0); forpart(q=n, s+=permcount(q)*Fix(q,x)); Vecrev(s/n!)}
{ for(n=0, 4, print(Row(n))) } \\ Andrew Howroyd, Mar 26 2020

T(n,k) = A371830(n,k-1) + A371830(n,k) (with A371830(n,k) = 0 if k < 0 or k >= 2^n). - Pontus von Brömssen, Apr 10 2024

A055609 Number of 3 X n binary matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 5, 17, 42, 91, 180, 328, 565, 930, 1470, 2248, 3344, 4849, 6881, 9579, 13104, 17649, 23442, 30736, 39833, 51074, 64842, 81574, 101766, 125959, 154771, 188883, 229044, 276085, 330926, 394558, 468083, 552696, 649692, 760482, 886602, 1029691, 1191539, 1374065, 1579326

Offset: 1

Vladeta Jovovic, Jun 03 2000

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1, -3, -1, 3, 6, -6, -3, 1, 3, 1, -3, 1).

Column k=3 of A056152.

Cf. A024206, A002623, A002727.

Vec((G(3, x) - G(2, x))*(1 - x) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: x*(x^8-x^7-x^6-2*x^5+2*x^4+x^3-3*x^2-2*x-1)/((x^3-1)^2*(x^2-1)^2*(x-1)^3).

Terms a(37) and beyond from Andrew Howroyd, Mar 25 2020

A052264 Number of 5 X n binary matrices up to row and column permutations.

Original entry on oeis.org

1, 6, 34, 190, 1053, 5624, 28576, 136758, 613894, 2583164, 10208743, 38013716, 133872584, 447620002, 1426354541, 4346885204, 12710830673, 35768703586, 97125981825, 255111287298, 649598148384, 1606754306778, 3867515638005, 9074220508038, 20784247213232

Offset: 0

Vladeta Jovovic, Feb 04 2000

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Binary matrices up to row and column permutations
Míšek [Misek], Bohuslav: O počtu tříd silně ekvivalentních incidenčních matic. (Czech) [On the number of classes of strongly equivalent incidence matrices]. Časopis pro pěstování matematiky, vol. 89 (1964), issue 2, pp. 211-218.
Index entries for linear recurrences with constant coefficients, signature (8, -24, 30, -6, -18, 27, -60, 87, -108, 147, -36, -82, 8, -147, 260, -253, 672, -413, -14, -471, -270, 612, -330, 2024, -1042, 213, -2022, -423, -18, 600, 4032, -858, 1468, -4952, -714, -3255, 1722, 5577, 1638, 4032, -5862, -1352, -8594, 1530, 3114, 5619, 6306, -2324, -170, -11814, -170, -2324, 6306, 5619, 3114, 1530, -8594, -1352, -5862, 4032, 1638, 5577, 1722, -3255, -714, -4952, 1468, -858, 4032, 600, -18, -423, -2022, 213, -1042, 2024, -330, 612, -270, -471, -14, -413, 672, -253, 260, -147, 8, -82, -36, 147, -108, 87, -60, 27, -18, -6, 30, -24, 8, -1).

Cf. A002623, A002727, A006148.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Vec(G(5, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^68 - 2*x^67 + 10*x^66 + 32*x^65 + 175*x^64 + 794*x^63 + 3441*x^62 + 13186*x^61 + 46027*x^60 + 146118*x^59 + 427347*x^58 + 1155432*x^57 + 2912873*x^56 + 6875608*x^55 + 15281029*x^54 + 32094658*x^53 + 63945531*x^52 + 121210914*x^51 + 219194198*x^50 + 378998758*x^49 + 627863648*x^48 + 998282344*x^47 + 1525746624*x^46 + 2244502676*x^45 + 3181886869*x^44 + 4351201210*x^43 + 5744918381*x^42 + 7328807372*x^41 + 9039504349*x^40 + 10785767638*x^39 + 12455264802*x^38 + 13925287384*x^37 + 15077477135*x^36 + 15812782150*x^35 + 16065602576*x^34 + 15812782150*x^33 + 15077477135*x^32 + 13925287384*x^31 + 12455264802*x^30 + 10785767638*x^29 + 9039504349*x^28 + 7328807372*x^27 + 5744918381*x^26 + 4351201210*x^25 + 3181886869*x^24 + 2244502676*x^23 + 1525746624*x^22 + 998282344*x^21 + 627863648*x^20 + 378998758*x^19 + 219194198*x^18 + 121210914*x^17 + 63945531*x^16 + 32094658*x^15 + 15281029*x^14 + 6875608*x^13 + 2912873*x^12 + 1155432*x^11 + 427347*x^10 + 146118*x^9 + 46027*x^8 + 13186*x^7 + 3441*x^6 + 794*x^5 + 175*x^4 + 32*x^3 + 10*x^2 - 2*x + 1)/((x^6 - 1)^2*(x^4 + x^3 + x^2 + x + 1)^6*(x^3 - x^2 + x - 1)^6 * (x^2 + x + 1)^6*(x + 1)^10*(x - 1)^24).

Name clarified by Ching Pong Siu, Aug 30 2022

A055066 Number of 7-covers of an unlabeled n-set.

Original entry on oeis.org

1, 7, 62, 664, 8609, 127415, 2004975, 31500927, 474504448, 6708348262, 88249739792, 1078567590128, 12269901302433, 130370516668917, 1298891291366245, 12182760243381355, 107979270564656625, 907568508195185203, 7256984238345563764, 55365443728411530716, 404091280028746802188

Offset: 0

Vladeta Jovovic, Jun 12 2000

nonn

Number of 7 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Binary matrices up to row and column permutations

Column 7 of A055080.

Cf. A005748, A002620, A005783, A005784, A005785, A005786.

Vec(G(7, x)*(1-x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

a(0)=1 prepended by Alois P. Heinz, Aug 08 2022

Terms a(17) and beyond from Andrew Howroyd, Feb 28 2023

A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 3, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 6, 11, 19, 10, 4, 1, 1, 1, 7, 16, 41, 32, 16, 4, 1, 1, 1, 8, 23, 81, 101, 68, 20, 5, 1, 1, 1, 9, 31, 153, 299, 301, 114, 29, 5, 1, 1, 1, 10, 41, 273, 849, 1358, 757, 210, 35, 6, 1

Offset: 0

Andrew Howroyd, May 28 2023

T(n,k) is also the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows and columns.

			Array begins:
======================================================
n/k| 0 1  2   3    4     5      6       7        8 ...
---+--------------------------------------------------
0  | 1 1  1   1    1     1      1       1        1 ...
1  | 1 1  1   1    1     1      1       1        1 ...
2  | 1 2  3   4    5     6      7       8        9 ...
3  | 1 2  4   7   11    16     23      31       41 ...
4  | 1 3  8  19   41    81    153     273      468 ...
5  | 1 3 10  32  101   299    849    2290     5901 ...
6  | 1 4 16  68  301  1358   6128   27008   114763 ...
7  | 1 4 20 114  757  5567  43534  343656  2645494 ...
8  | 1 5 29 210 1981 23350 319119 4633380 67013431 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)

A259344 is the same array without the first row and column read by upward antidiagonals.
Columns k=0..6 are A000012, A004526(n+2), A005232, A006381, A006382, A056204, A056205.
Rows n=2..4 are A000027(n+1), A000601, A006380.
Main diagonal is A006383.
Cf. A028657, A241956, A362905.

\\ Compare A028657.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={sum(j=1, #q, gcd(t, q[j]))}
T(n, k)={if(n==0, 1, my(s=0); forpart(q=n, my(e=1<

A005784 Number of 4-covers of an unlabeled n-set.

Original entry on oeis.org

1, 4, 17, 65, 230, 736, 2197, 6093, 15864, 38960, 90837, 202005, 430577, 883057, 1748909, 3355213, 6252575, 11345602, 20089514, 34778306, 58964020, 98053576, 160151566, 257229974, 406739271, 633795181, 974126408, 1477999320, 2215409037, 3282874359, 4812278064

Offset: 0

N. J. A. Sloane

Number of 4 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stefano Spezia, Table of n, a(n) for n = 0..10000
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Vladeta Jovovic, Binary matrices up to row and column permutations
Index entries for linear recurrences with constant coefficients, signature (5, -7, -1, 5, -1, 21, -33, 0, -4, 8, 68, -57, -3, -57, 13, 100, -32, 32, -100, -13, 57, 3, 57, -68, -8, 4, 0, 33, -21, 1, -5, 1, 7, -5, 1).

Column 4 of A055080.
First differences of A006148.
Cf. A002620, A005783, A005785, A005786, A055066.

Vec(G(4, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^20 - x^19 + 4*x^18 + 9*x^17 + 23*x^16 + 39*x^15 + 90*x^14 + 131*x^13 + 204*x^12 + 238*x^11 + 252*x^10 + 238*x^9 + 204*x^8 + 131*x^7 + 90*x^6 + 39*x^5 + 23*x^4 + 9*x^3 + 4*x^2 - x + 1)/((1 - x^4)^3*(1 - x^3)^4*(1 - x^2)^3*(1 - x)^5).
a(n) ~ n^14/2092278988800. - Stefano Spezia, Aug 08 2022
a(n) = n^14/2092278988800 + n^13/19926466560 + n^12/418037760 + n^11/14598144 + 689*n^10/522547200 + 253*n^9/13934592 + 2184839*n^8/11705057280 + 10313*n^7/6967296 + 2319707*n^6/250822656 + 1817221*n^5/39813120 + 2405336243*n^4/13795246080 + 151784975*n^3/306561024 + 93746545019*n^2/95103590400 + 924100468541*n/717352796160 + 1 + (n^2/486 + 5*n/162 + 233/2187)*floor(n/3) + (n^2/256 + 15*n/256 + 101/512)*floor(n/4) - (n^3/1458 + 7*n^2/486 + 22*n/243 + 356/2187)*floor((n+1)/3) + (n^5/122880 + 5*n^4/16384 + 125*n^3/24576 + 359*n^2/8192 + 10967*n/61440 + 8461/32768)*floor(n/2) + (n/256 + 15/512)*floor((n+1)/4). - Vaclav Kotesovec, Aug 09 2022

More terms from Vladeta Jovovic, Jun 03 2000

a(0)=1 prepended by Alois P. Heinz, Aug 08 2022

A005785 Number of 5-covers of an unlabeled n-set.

Original entry on oeis.org

1, 5, 28, 156, 863, 4571, 22952, 108182, 477136, 1969270, 7625579, 27804973, 95858868, 313747418, 978734539, 2920530663, 8363945469, 23057872913, 61357278239, 157985305473, 394486861086, 957156158394, 2260761331227

Offset: 0

N. J. A. Sloane

Number of 5 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stefano Spezia, Table of n, a(n) for n = 0..10000
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Vladeta Jovovic, Binary matrices up to row and column permutations
Index entries for linear recurrences with constant coefficients, signature (7, -17, 13, 7, -11, 16, -44, 43, -65, 82, 46, -36, -28, -175, 85, -168, 504, 91, 77, -394, -664, -52, -382, 1642, 600, 813, -1209, -1632, -1650, -1050, 2982, 2124, 3592, -1360, -2074, -5329, -3607, 1970, 3608, 7640, 1778, 426, -8168, -6638, -3524, 2095, 8401, 6077, 5907, -5907, -6077, -8401, -2095, 3524, 6638, 8168, -426, -1778, -7640, -3608, -1970, 3607, 5329, 2074, 1360, -3592, -2124, -2982, 1050, 1650, 1632, 1209, -813, -600, -1642, 382, 52, 664, 394, -77, -91, -504, 168, -85, 175, 28, 36, -46, -82, 65, -43, 44, -16, 11, -7, -13, 17, -7, 1).

Column 5 of A055080.
First differences of A052264.
Cf. A002620, A005783, A005784, A055066.

Vec(G(5, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: - (x^68 - 2*x^67 + 10*x^66 + 32*x^65 + 175*x^64 + 794*x^63 + 3441*x^62 + 13186*x^61 + 46027*x^60 + 146118*x^59 + 427347*x^58 + 1155432*x^57 + 2912873*x^56 + 6875608*x^55 + 15281029*x^54 + 32094658*x^53 + 63945531*x^52 + 121210914*x^51 + 219194198*x^50 + 378998758*x^49 + 627863648*x^48 + 998282344*x^47 + 1525746624*x^46 + 2244502676*x^45 + 3181886869*x^44 + 4351201210*x^43 + 5744918381*x^42 + 7328807372*x^41 + 9039504349*x^40 + 10785767638*x^39 + 12455264802*x^38 + 13925287384*x^37 + 15077477135*x^36 + 15812782150*x^35 + 16065602576*x^34 + 15812782150*x^33 + 15077477135*x^32 + 13925287384*x^31 + 12455264802*x^30 + 10785767638*x^29 + 9039504349*x^28 + 7328807372*x^27 + 5744918381*x^26 + 4351201210*x^25 + 3181886869*x^24 + 2244502676*x^23 + 1525746624*x^22 + 998282344*x^21 + 627863648*x^20 + 378998758*x^19 + 219194198*x^18 + 121210914*x^17 + 63945531*x^16 + 32094658*x^15 + 15281029*x^14 + 6875608*x^13 + 2912873*x^12 + 1155432*x^11 + 427347*x^10 + 146118*x^9 + 46027*x^8 + 13186*x^7 + 3441*x^6 + 794*x^5 + 175*x^4 + 32*x^3 + 10*x^2 - 2*x + 1)/((x^6 - 1)^2*(x^4 + x^3 + x^2 + x + 1)^6*(x^3 - x^2 + x - 1)^6*(x^2 + x + 1)^6*(x + 1)^10*(x - 1)^23).

a(n) = n^30/(30!*5!) + O(n^29). - Vaclav Kotesovec, Aug 09 2022

More terms from Vladeta Jovovic, Jun 03 2000

a(0) = 1 prepended by Stefano Spezia, Aug 09 2022

A005786 Number of 6-covers of an unlabeled n-set.

Original entry on oeis.org

1, 6, 43, 336, 2864, 25326, 223034, 1890123, 15115098, 112980937, 787320629, 5121184083, 31188412225, 178517111561, 964196387369, 4933278065881, 23997707450765, 111358094980387, 494444748602595, 2106504840061571

Offset: 0

N. J. A. Sloane

nonn

Number of 6 X n binary matrices with at least one 1 in every column up to row and column permutations.

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Vladeta Jovovic, Binary matrices up to row and column permutations

Column 6 of A055080.

Cf. A002620, A005783, A005784, A005785, A055066.

Vec(G(6, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

More terms from Vladeta Jovovic, Jun 12 2000

a(0)=1 prepended by Alois P. Heinz, Aug 08 2022

A056152 Triangular array giving number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 8, 17, 8, 1, 1, 11, 42, 42, 11, 1, 1, 15, 91, 179, 91, 15, 1, 1, 19, 180, 633, 633, 180, 19, 1, 1, 24, 328, 2001, 3835, 2001, 328, 24, 1, 1, 29, 565, 5745, 20755, 20755, 5745, 565, 29, 1, 1, 35, 930, 15274, 102089, 200082, 102089

Offset: 2

Vladeta Jovovic, Jul 29 2000

Also table read by rows: for 0 < k < n, a(n, k) = number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k vertices, up to isomorphism.
a(n, k) is the number of isomorphism classes of finite subdirectly irreducible almost distributive lattices in which the N-quotient has k upper covers and (n - k) lower covers. - David Wasserman, Feb 11 2002
Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color, with no isolated nodes; the color classes are not interchangeable.

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  8,  17,   8,  1;
  1, 11,  42,  42,  11,  1;
  1, 15,  91, 179,  91,  15,  1;
  1, 19, 180, 633, 633, 180, 19, 1;
  ...
There are 17 bipartite graphs with 6 vertices, no isolated vertices and a distinguished bipartite block with 3 vertices, or equivalently, there are 17 3 X 3 binary matrices with no zero rows or columns, up to row and column permutation:
[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
[0 0 1] [0 0 1] [0 1 0] [0 1 0] [0 1 0] [0 1 1] [0 1 1] [0 1 1] [1 1 0]
[1 1 0] [1 1 1] [1 0 0] [1 0 1] [1 1 1] [1 0 1] [1 1 0] [1 1 1] [1 1 0]
and
[0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
[1 1 0] [1 1 1] [0 1 1] [0 1 1] [1 0 1] [1 0 1] [1 1 1] [1 1 1]
[1 1 1] [1 1 1] [1 0 1] [1 1 1] [1 1 0] [1 1 1] [1 1 1] [1 1 1].

J. G. Lee, Almost Distributive Lattice Varieties, Algebra Universalis, 21 (1985), 280-304.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43. See Table 2.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 2.

Columns k=1..6 are A000012, A024206, A055609, A055082, A055083, A055084.
Row sums give A055192.
See A122083 for another version of this triangle.
Cf. A049312, A048194, A028657, A049311.

A002725 Number of incidence matrices: n X (n+1) binary matrices under row and column permutations.

Original entry on oeis.org

1, 3, 13, 87, 1053, 28576, 2141733, 508147108, 402135275365, 1073376057490373, 9700385489355970183, 298434346895322960005291, 31479360095907908092817694945, 11474377948948020660089085281068730, 14568098446466140788730090352230460100956

Offset: 0

N. J. A. Sloane

a(0) = 1 by convention.

			a(1) = 3: [0,0], [0,1], [1,1].
a(2) = 13:
000 000 000 000 001 001 001 001 001 011 011 011 111
000 001 011 111 001 010 011 110 111 011 101 111 111

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).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..23 from Alois P. Heinz)
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.

Cf. A002623, A002727, A006148, A002728, A002724.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> add(add(2^add(add(igcd(i, j)* coeff(s, x, i)*
      coeff(t, x, j), j=1..degree(t)), i=1..degree(s))/
      mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))/
      mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t)),
      t=b(n+1$2)), s=b(n$2)):
seq(a(n), n=0..12);  # Alois P. Heinz, Aug 01 2014

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten @ Table[ Map[ Function[ {p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}]]];
a[n_] := Sum[Sum[2^Sum[ Sum [ GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[ i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+1, n+1]}], {s,  b[n, n]}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

a(n) = A(n+1,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

a(n) = sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n+1} (fix A[s_1, s_2, ...; t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fix A[...] = 2^sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Sean A. Irvine, Jul 31 2014

More terms from Vladeta Jovovic, Feb 04 2000

cp's OEIS Frontend

A052265 Triangle giving T(n,r) = number of equivalence classes of Boolean functions of n variables and range r=0..2^n under action of symmetric group.

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A055609 Number of 3 X n binary matrices with no zero rows or columns, up to row and column permutation.

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A052264 Number of 5 X n binary matrices up to row and column permutations.

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A055066 Number of 7-covers of an unlabeled n-set.

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A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

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A005784 Number of 4-covers of an unlabeled n-set.

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A005785 Number of 5-covers of an unlabeled n-set.

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A005786 Number of 6-covers of an unlabeled n-set.

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