A318813 Number of balanced reduced multisystems with n atoms all equal to 1.

1, 1, 2, 6, 20, 90, 468, 2910, 20644, 165874, 1484344, 14653890, 158136988, 1852077284, 23394406084, 317018563806, 4587391330992, 70598570456104, 1151382852200680, 19835976878704628, 359963038816096924, 6863033015330999110, 137156667020252478684, 2867083618970831936826

Offset

1,3

Comments

For n > 1, also the number of balanced reduced multisystems whose atoms are an integer partition of n with at least one part > 1. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. - Gus Wiseman, Dec 31 2019

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Formula

a(n > 1) = A330679(n)/2. - Gus Wiseman, Dec 31 2019

Example

The a(5) = 20 balanced reduced multisystems (with n written in place of 1^n):
  5  (14)  (23)  (113)      (122)      (1112)
                 ((1)(13))  ((1)(22))  ((1)(112))
                 ((3)(11))  ((2)(12))  ((2)(111))
                                       ((11)(12))
                                       ((1)(1)(12))
                                       ((1)(2)(11))
                                       (((1))((1)(12)))
                                       (((1))((2)(11)))
                                       (((2))((1)(11)))
                                       (((12))((1)(1)))
                                       (((11))((1)(2)))

Mathematica

normize[m_]:=m/.Rule@@@Table[{Union[m][[i]], i}, {i, Length[Union[m]]}];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
totfact[n_]:=totfact[n]=1+Sum[totfact[Times@@Prime/@normize[f]], {f, Select[facs[n], 1<Length[#]<PrimeOmega[n]&]}];
Table[totfact[2^n], {n, 10}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n, i, i==1), u=vector(n)); for(r=1, #v, u += v*sum(j=r, #v, (-1)^(j-r)*binomial(j-1, r-1)); v=EulerT(v)); u} \\ Andrew Howroyd, Dec 30 2019

Crossrefs

The maximum-depth case is A000111.

Cf. A000311, A001055, A002846, A005121, A213427, A281118, A281119, A317145, A318812, A318846, A320154, A330474, A330679.

Sequence in context: A003069 A079468 A124382 * A336544 A000666 A180890

Adjacent sequences: A318810 A318811 A318812 * A318814 A318815 A318816

Keyword

nonn

Author

Gus Wiseman, Sep 04 2018

Extensions

Terms a(14) and beyond from Andrew Howroyd, Dec 30 2019

Terminology corrected by Gus Wiseman, Dec 31 2019

Status

approved