A007716 -id:A007716 - cp's OEIS frontend

A321390 Third Moebius transform of A007716. Number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods and whose dual is also an aperiodic multiset partition.

1, 1, 1, 7, 24, 88, 265, 907, 2929, 9918, 33931, 119366, 428314, 1574221, 5913415, 22699536, 88994103, 356058537, 1453049451, 6044132791, 25612496016, 110503624870, 485160989937, 2166488899639, 9835208617114, 45370059225048

Offset: 0

Gus Wiseman, Nov 08 2018

nonn

The Moebius transform c of a sequence b is c(n) = Sum_{d|n} mu(d) * b(n/d).
Also the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns where the multiset of rows and the multiset of columns are both aperiodic and the nonzero entries are relatively prime, up to row and column permutations.
A multiset is aperiodic if its multiplicities are relatively prime. The period of a multiset is the GCD of its multiplicities.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 24 multiset partitions:
  {{1}}  {{1},{2}}  {{1,2,2}}      {{1,2,2,2}}
                    {{1},{1,1}}    {{1,2,3,3}}
                    {{1},{2,2}}    {{1},{1,1,1}}
                    {{1},{2,3}}    {{1},{1,2,2}}
                    {{2},{1,2}}    {{1},{2,2,2}}
                    {{1},{2},{2}}  {{1,2},{2,2}}
                    {{1},{2},{3}}  {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320800-A320810.

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

Original entry on oeis.org

1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876, 203286806, 340435588, 568496753, 946695386

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g., let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.
Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g., for n = 4 we have:
  4   31   3   22   2   211   21   2    2   1111   111   11    11   1
           1        2         1    11   1          1     11    1    1
                                        1                      1    1
                                                                    1
Also total number of different species of singularity for conjugate functions with n letters (Sylvester).
According to [Belmans], this sequence gives "[t]he number of Segre symbols for the intersection of two quadrics in a fixed dimension". - Eric M. Schmidt, Sep 02 2017
From Gus Wiseman, Jul 30 2022: (Start)
Also the number of non-isomorphic multiset partitions of weight n with all constant blocks. The strict case is A089259. For example, non-isomorphic representatives of the a(1) = 1 through a(3) = 6 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{1},{1},{1}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
A000688 counts factorizations into prime powers.
A007716 counts non-isomorphic multiset partitions by weight.
A279784 counts twice-partitions of type PPR, factorizations A295935.
Constant partitions are ranked by prime-powers: A000961, A023894, A054685, A246655, A355743.
(End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + ...
a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).
The a(4)=14 multiset partitions whose total sum of parts is 4 are:
((4)),
((13)), ((1)(3)),
((22)), ((2)(2)),
((112)), ((1)(12)), ((2)(11)), ((1)(1)(2)),
((1111)), ((1)(111)), ((11)(11)), ((1)(1)(11)), ((1)(1)(1)(1)). - _Gus Wiseman_, Dec 19 2016

A. Cayley, Recherches sur les matrices dont les termes sont des fonctions linéaires d'une seule indéterminée, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
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).
J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order', Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 (first 500 terms from T. D. Noe)
Pieter Belmans, Segre symbols, 2016.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See pp. 10, 16.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).
L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, 2014.
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, Science China Physics, Mechanics & Astronomy, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9.
Jessie Pitsillides, Segre Characteristic Equivalence, arXiv:2506.12065 [math.GM], 2025.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
N. J. A. Sloane, Transforms.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
Index entries for sequences related to rooted trees

Related to A001383 via generating function.
The multiplicative version (factorizations) is A050336.
The ordered version (sequences of partitions) is A055887.
Row-sums of A061260.
Main diagonal of A055885.
We have A271619(n) <= a(n) <= A063834(n).
Column k=3 of A290353.
The strict case is A316980.
Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
Cf. A089300.
Cf. A001055, A072233, A112798, A275024.

Following Vladeta Jovovic:
a001970 n = a001970_list !! (n-1)
a001970_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y : ys) where
            y = sum (zipWith (*) ys a061259_list) `div` x
-- Reinhard Zumkeller, Oct 31 2015

with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
# second Maple program:
with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 19 2016

m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* Jean-François Alcover, Jul 19 2011, after g.f. *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k + x * O(x^n)), n))}; /* Michael Somos, Dec 20 2016 */

from sympy.core.cache import cacheit
from sympy import npartitions, divisors
@cacheit
def a(n): return 1 if n == 0 else sum([sum([d*npartitions(d) for d in divisors(j)])*a(n - j) for j in range(1, n + 1)]) / n
[a(n) for n in range(51)]  # Indranil Ghosh, Aug 19 2017, after Maple code
# (Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 1)
a = EulerTransform(EulerTransform(b))
print([a(n) for n in range(36)]) # Peter Luschny, Nov 17 2022

G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]
a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n > 1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic, Apr 21 2001
Logarithmic derivative yields A061259 (equivalent to above formula from Vladeta Jovovic). - Paul D. Hanna, Sep 05 2012
a(n) = Sum_{k=1..A000041(n)} A001055(A215366(n,k)) = number of factorizations of Heinz numbers of integer partitions of n. - Gus Wiseman, Dec 19 2016
a(n) = |{m>=1 : n = Sum_{k=1..A001222(m)} A056239(A112798(m,k)+1)}| = number of normalized twice-prime-factored multiset partitions (see A275024) whose total sum of parts is n. - Gus Wiseman, Dec 19 2016

Additional comments from Valery A. Liskovets

Sylvester references from Barry Cipra, Oct 07 2003

A302242 Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 2, 2, 2, 0, 1, 2, 3, 1, 3, 1, 2, 1, 2, 2, 3, 2, 2, 2, 1, 0, 2, 1, 3, 2, 3, 3, 3, 1, 1, 3, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 4, 3, 2, 2, 4, 2, 1, 2, 3, 1, 4, 0, 3, 2, 1, 1, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 1, 4, 1, 1, 3, 2, 2, 3, 1, 4

Offset: 1

Gus Wiseman, Apr 03 2018

nonn

A multiset multisystem is a finite multiset of finite multisets of positive integers. The n-th multiset multisystem is constructed by factoring n into prime numbers and then factoring each prime index into prime numbers and taking their prime indices. This produces a unique multiset multisystem for each n, and every possible multiset multisystem is so constructed as n ranges over all positive integers.

			Sequence of finite multisets of finite multisets of positive integers begins: (), (()), ((1)), (()()), ((2)), (()(1)), ((11)), (()()()), ((1)(1)), (()(2)), ((3)), (()()(1)), ((12)), (()(11)), ((1)(2)), (()()()()), ((4)), (()(1)(1)), ((111)), (()()(2)).

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A001222, A003963, A007716, A034691, A056239, A061775, A063834, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302243.

with(numtheory):
a:= n-> add(add(j[2], j=ifactors(pi(i[1]))[2])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 07 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[PrimeOmega/@primeMS[n]],{n,100}]

a(n,f=factor(n))=sum(i=1,#f~, bigomega(primepi(f[i,1]))*f[i,2]) \\ Charles R Greathouse IV, Nov 10 2021

A283877 Number of non-isomorphic set-systems of weight n.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 44, 98, 244, 605, 1595, 4273, 12048, 34790, 104480, 322954, 1031556, 3389413, 11464454, 39820812, 141962355, 518663683, 1940341269, 7424565391, 29033121685, 115921101414, 472219204088, 1961177127371, 8298334192288, 35751364047676, 156736154469354

Offset: 0

Gus Wiseman, Mar 17 2017

nonn

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.

			Non-isomorphic representatives of the a(4)=9 set-systems are:
((1234)),
((1)(234)), ((3)(123)), ((12)(34)), ((13)(23)),
((1)(2)(12)), ((1)(2)(34)), ((1)(3)(23)),
((1)(2)(3)(4)).

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A034691, A049311, A056156, A089259, A116540, A300913.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024

Euler transform of A300913.

a(0) = 1 prepended and terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

Original entry on oeis.org

1, 3, 6, 16, 34, 90, 211, 558, 1430, 3908, 10725, 30825, 90156, 273234, 848355, 2714399, 8909057, 30042866, 103859678, 368075596, 1335537312, 4958599228, 18820993913, 72980867400, 288885080660, 1166541823566, 4802259167367, 20141650236664

Offset: 1

Peter J. Cameron

Also the number of bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism.
The EULERi transform (A056156) is also interesting.
a(n) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n. - Gus Wiseman, Mar 17 2017

			E.g. a(2) = 3: two ones in same row, two ones in same column, or neither.
a(3) = 6 is coefficient of x^3 in (1/36)*((1 + x)^9 + 6*(1 + x)^3*(1 + x^2)^3 + 8*(1 + x^3)^3 + 9*(1 + x)*(1 + x^2)^4 + 12*(1 + x^3)*(1 + x^6))=1 + x + 3*x^2 + 6*x^3 + 7*x^4 + 7*x^5 + 6*x^6 + 3*x^7 + x^8 + x^9.
There are a(3) = 6 binary matrices with 3 ones, with no zero rows or columns, up to row and column permutation:
  [1 0 0] [1 1 0] [1 0] [1 1] [1 1 1] [1]
  [0 1 0] [0 0 1] [1 0] [1 0] ....... [1].
  [0 0 1] ....... [0 1] ............. [1]
Non-isomorphic representatives of the a(3)=6 set multipartitions are: ((123)), ((1)(23)), ((2)(12)), ((1)(1)(1)), ((1)(2)(2)), ((1)(2)(3)). - _Gus Wiseman_, Mar 17 2017

Aliaksandr Siarhei, Table of n, a(n) for n = 1..102
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
Peter J. Cameron, Problems on Permutation Groups, see Problem 3
Index entries for sequences related to binary matrices.

Main diagonal of A321609.
Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724.
Cf. A057149, A057150, A057151, A057152.
Cf. A034691, A056156, A089259, A116540, A283877.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
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, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 16 2023

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, then apply EULER transform.

a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; 1+x, 1+x^2, ...), where S_n X S_n is Cartesian product of symmetric groups S_n of degree n.

More terms and formula from Vladeta Jovovic, Jul 29 2000
a(19)-a(28) from Max Alekseyev, Jul 22 2009
a(29)-a(102) from Aliaksandr Siarhei, Dec 13 2013
Name edited by Gus Wiseman, Dec 18 2018

A000612 Number of P-equivalence classes of switching functions of n or fewer variables, divided by 2.

Original entry on oeis.org

1, 2, 6, 40, 1992, 18666624, 12813206169137152, 33758171486592987164087845043830784, 1435913805026242504952006868879460423834904914948818373264705576411070464

Offset: 0

N. J. A. Sloane

Also number of nonisomorphic sets of nonempty subsets of an n-set.
Equivalently, number of nonisomorphic fillings of a Venn diagram of n sets. - Joerg Arndt, Mar 24 2020
Number of hypergraphs on n unlabeled nodes. - Charles R Greathouse IV, Apr 06 2021

			Non-isomorphic representatives of the a(2) = 6 set-systems are 0, {1}, {12}, {1}{2}, {1}{12}, {1}{2}{12}. - _Gus Wiseman_, Aug 07 2018

M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 153.
S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 Table 2.3.2. - Row 5.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..12
M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561.
M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12 (1963), 559-561. [Annotated scanned copy]
Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, Hypergraph Motifs and Their Extensions Beyond Binary, arXiv:2310.15668 [cs.SI], 2023.
Peter H. van der Kamp, Hypergraphs and homogeneous Lotka-Volterra systems with linear Darboux polynomials, arXiv:2411.18264 [nlin.SI], 2024. See p. 4.
Wikipedia, Venn diagram
Index entries for sequences related to Boolean functions

a(n) = A003180(n)/2.

Cf. A007716, A055621, A058891, A283877, A300913, A306005, A317533, A317757.

a:= n-> add(1/(p-> mul((c-> j^c*c!)(coeff(p, x, j)), j=1..degree(p)))(
        add(x^i, i=l))*2^((w-> add(mul(2^igcd(t, l[i]), i=1..nops(l)),
        t=1..w)/w)(ilcm(l[]))), l=combinat[partition](n))/2:
seq(a(n), n=0..9);  # Alois P. Heinz, Aug 12 2019

sysnorm[{}] := {};sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Subsets[Rest[Subsets[Range[n]]]]]],{n,4}] (* Gus Wiseman, Aug 07 2018 *)
a[n_] := Sum[1/Function[p, Product[Function[c, j^c*c!][Coefficient[p, x, j]], {j, 1, Exponent[p, x]}]][Total[x^l]]*2^(Function[w, Sum[Product[2^GCD[t, l[[i]]], {i, 1, Length[l]}], {t, 1, w}]/w][If[l=={}, 1, LCM @@ l]]), {l, IntegerPartitions[n]}]/2;
a /@ Range[0, 9] (* Jean-François Alcover, Feb 04 2020, after Alois P. Heinz *)

def partition(n, I=1):
  yield () if n==0 else (n,)
  for i in range(I, n//2 + 1):
    for p in partition(n-i, i):
      yield (i,) + p
def a(n):
  import math, operator, functools
  fracs = [(1<<(sum(functools.reduce(operator.mul, (1<Gregory Morse, Dec 23 2024

a(n) = A003180(n)/2.

More terms from Vladeta Jovovic, Feb 23 2000

A007718 Number of independent polynomial invariants of matrix of order n.

Original entry on oeis.org

1, 1, 3, 6, 17, 40, 125, 354, 1159, 3774, 13113, 46426, 171027, 644038, 2493848, 9867688, 39922991, 164747459, 693093407, 2968918400, 12940917244, 57353242370, 258306634422, 1181572250326, 5486982683756, 25856584485254

Offset: 0

Colin Mallows

nonn

Also the number of non-isomorphic connected multiset partitions of weight n. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices. - Gus Wiseman, Sep 23 2018

			From _Gus Wiseman_, Sep 24 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(4) = 17 connected multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
        {{1},{1}}   {{1,2,3}}      {{1,2,2,2}}
                   {{1},{1,1}}     {{1,2,3,3}}
                   {{2},{1,2}}     {{1,2,3,4}}
                  {{1},{1},{1}}   {{1},{1,1,1}}
                                  {{1},{1,2,2}}
                                  {{2},{1,2,2}}
                                  {{3},{1,2,3}}
                                  {{1,1},{1,1}}
                                  {{1,2},{1,2}}
                                  {{1,2},{2,2}}
                                  {{1,3},{2,3}}
                                 {{1},{1},{1,1}}
                                 {{1},{2},{1,2}}
                                 {{2},{2},{1,2}}
                                {{1},{1},{1},{1}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A007718, A056156, A319557, A319565, A319566.

Inverse Euler transform of A007716.

a(7)-a(25) from Franklin T. Adams-Watters, Jun 21 2011

a(0)=1 prepended by Andrew Howroyd, Jan 15 2023

A316980 Number of non-isomorphic strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 3, 8, 23, 63, 197, 588, 1892, 6140, 20734, 71472, 254090, 923900, 3446572, 13149295, 51316445, 204556612, 832467052, 3455533022, 14621598811, 63023667027, 276559371189, 1234802595648, 5606647482646, 25875459311317, 121324797470067, 577692044073205

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows (or alternatively, with no equal columns).

Also the number of non-isomorphic multiset partitions of weight n with no equivalent vertices. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 8 multiset partitions with no equivalent vertices (first column) and with no equal blocks (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(1)(1) <-> (123)
  (1)(2)(2) <-> (1)(23)
  (1)(2)(3) <-> (1)(2)(3)

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091.
Cf. A316974, A316978, A316979, A316981, A316983.
Cf. A049311, A059201, A319558, A319560, A319567.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(p=sum(t=1, n, subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*polcoef(exp(p-subst(p,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 21 2023

Euler transform of A319557. - Gus Wiseman, Sep 23 2018

a(7)-a(10) from Gus Wiseman, Sep 23 2018

Terms a(11) and beyond from Andrew Howroyd, Jan 19 2023

A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 9, 17, 36, 72, 155, 319, 677, 1429, 3094, 6648, 14518, 31796, 70491, 156818, 352371, 795952, 1813580, 4155367, 9594425, 22283566, 52122379, 122631874, 290432439, 691831161, 1658270316, 3997272089, 9692519896, 23631827354, 57943821449, 142834652193

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n, under row and column permutations.

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity.

			Non-isomorphic representatives of the a(4) = 9 self-dual multiset partitions:
  (1111),
  (1)(222), (2)(122), (11)(22), (12)(12),
  (1)(1)(23), (1)(2)(33), (1)(3)(23),
  (1)(2)(3)(4).
The a(4) = 9 square symmetric matrices:
. [4]
.
. [3 0]  [2 0]  [2 1]  [1 1]
. [0 1]  [0 2]  [1 0]  [1 1]
.
. [2 0 0]  [1 1 0]  [0 1 1]
. [0 1 0]  [1 0 0]  [1 0 0]
. [0 0 1]  [0 0 1]  [1 0 0]
.
. [1 0 0 0]
. [0 1 0 0]
. [0 0 1 0]
. [0 0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A320796.
Main diagonal of A318805.
Cf. A000009, A001055, A007716, A007717, A020555, A045778.
Cf. A316974, A316978, A316979, A316980, A316981.

vector(25, n, n--; T(n,n)) \\ T(n,k) defined in A318805. - Andrew Howroyd, Jan 16 2024

Terms a(9) and beyond from Andrew Howroyd, Sep 03 2018

A255906 Number of collections of nonempty multisets with a total of n objects having color set {1,...,k} for some k<=n.

Original entry on oeis.org

1, 1, 4, 16, 76, 400, 2356, 15200, 106644, 806320, 6526580, 56231024, 513207740, 4941362512, 50013751812, 530481210672, 5880285873060, 67954587978448, 816935340368068, 10196643652651664, 131904973822724540, 1765645473517011568, 24420203895517396180

Offset: 0

Alois P. Heinz, Mar 10 2015

nonn

Number of multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Jul 30 2018

			a(0) = 1: {}.
a(1) = 1: {{1}}.
a(2) = 4: {{1},{1}}, {{1,1}}, {{1},{2}}, {{1,2}}.
a(3) = 16: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{1},{2}}, {{1},{2},{2}}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}, {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row sums of A255903. Also row sums of A317532.

Cf. A007716, A034691, A116539, A317073.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Join@@mps/@allnorm[n]],{n,6}] (* Gus Wiseman, Jul 30 2018 *)

R(n, k)={Vec(-1 + 1/prod(j=1, n, (1 - x^j + O(x*x^n))^binomial(k+j-1, j) ))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

a(n) = Sum_{k=0..n} A255903(n,k).

cp's OEIS Frontend

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A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

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A283877 Number of non-isomorphic set-systems of weight n.

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A007718 Number of independent polynomial invariants of matrix of order n.

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