OFFSET
0,3
COMMENTS
From Yifan Xie, Sep 25 2024: (Start)
a(n) is the number of distinct sets A = {b_1, b_2, ..., b_n} such that 2*n positive integers x_1, x_2, ..., x_(2*n) exist where A = {x_1+x_2, x_3+x_4, ..., x_(2*n-1)+x_(2*n)} = {x_1*x_2, x_3*x_4, ..., x_(2*n-1)*x_(2*n)}.
Proof: Suppose that the number of sets A is b(n). Denote (x_(2*i-1)-1)*(x_(2*i)-1) by c_i. (1 <= i <= n)
Taking the sums of B and C, c_1 + c_2 + ... + c_n = n. (1)
Consider b_1, ..., b_n as n vertices, then the map B -> C is a directed graph G on these vertices, where each vertex has a source and a sink, so it can either be a cycle itself or decomposed into two or more cycles.
For the first case, the condition is equivalent to every proper subset of A' = {b_1, ..., b_n} is invalid for the corresponding n. Using (1), every partial sum of c_i is not equal to the number of addends. Therefore, c_i != 1. Then there must exist c_j = 0, hence c_i != 2. Then there must exist another c_k = 0, hence c_i != 3, and so on. Thus c_1, c_2, ..., c_n must be a permutation of 0, 0, ..., 0, n. Suppose that c_n = n, x_1 = x_3 = ... = x_(2*n-3) = 1. Since n has floor((A000005(n)+1)/2) ways to be expressed as the product of two positive integers, each product n = y*z means that x_(2*i-1) = y+1, x_(2*i) = z+1, thus there exists (y+1)*(z+1) in A, 1 + x_(2*l) = (a+1)*(b+1), 1*x_(2*l) = (a+1)*(b+1)-1, there exists (a+1)*(b+1)-1 in A, and so on until a+b+2 = (a+1)*(b+1)-1. In conclusion, there are floor((A000005(n)+1)/2) distinct A's in the second case, each of which is a group of consecutive integers. Denote the array by n = a*b .
For the second case, the array A can be decomposed into smaller arrays representing smaller n's, without breaking the structures of B and C. This process will finally end with all smaller arrays in the first case. Using the same notation, the arrays can be expressed as n = y_1*z_1 + y_2*z_2 + ... + y_s*z*s.
Therefore b(n) is the number of representations of n as a sum of products of pairs of unordered positive integers, hence b(n) = a(n). (End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
FORMULA
Euler transform of A038548.
G.f.: Product_{k>0} 1/(1-x^k)^A038548(k).
G.f.: Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))). - Vaclav Kotesovec, Aug 19 2019
EXAMPLE
a(0) = 1: 0 = the empty sum.
a(1) = 1: 1 = 1*1.
a(2) = 2: 2 = 1*1 + 1*1 = 1*2.
a(3) = 3: 3 = 1*1 + 1*1 + 1*1 = 1*1 + 1*2 = 1*3.
a(4) = 6: 4 = 1*1 + 1*1 + 1*1 + 1*1 = 1*1 + 1*1 + 1*2 = 1*1 + 1*3 = 1*2 + 1*2 = 2*2 = 1*4.
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*ceil(tau(d)/2), d=divisors(j)) *a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..60);
MATHEMATICA
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Ceiling[DivisorSigma[0, d]/2], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[Product[1/(1 - x^(k*j)), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 22 2012
STATUS
approved