OFFSET
0,3
COMMENTS
From Gary W. Adamson, Feb 11 2010: (Start)
Given A000041, P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...),
A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...),
A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. (End)
Let M = triangle A173238 as an infinite lower triangular matrix. Then A092119 = lim_{n->infinity} M^n. Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + ...), and A(x) = polcoeff A092119. Then P(x) = A(x) / A(x^2), an example of a conjectured infinite set of operations (cf. A173238). - Gary W. Adamson, Feb 13 2010
a(n) is the number of nonzero terms appearing in the unsigned elementary polysymmetric function expansion of the signed elementary polysymmetric function E_n. - Aditya Khanna, Dec 03 2025
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Aditya Khanna, Transition Matrices between Plethystic Bases of Polysymmetric Functions via Bijective Methods, arXiv:2510.12723 [math.CO], 2025. See pp. 5, 46.
N. J. A. Sloane, Transforms
FORMULA
G.f.: 1/Product_{k>=0} P(x^(2^k)) where P(x) = Product_{k>=1} (1 - x^k). - Joerg Arndt, Jun 21 2011
a(n) = Sum_{P} Product_{i>=1} A000041(m_i(P)), where the sum is over binary partitions P of n (cf. A018819) and where m_i(P) is the multiplicity of part i in the partition P. - Aditya Khanna, Dec 05 2025
log(a(n)) ~ 2*Pi*sqrt(n/3). - Vaclav Kotesovec, Feb 21 2026
EXAMPLE
The binary partitions of 5 written in exponential notation are [1^5], [2^1, 1^3], [2^2, 1^1] [4^1, 1^1]. We compute a(5) using the explicit formula above as p(5) + p(1)*p(3) + p(2)*p(1) + p(1)*p(1) = 1 + 2 + 3 + 7 = 13, where p(n) = A000041(n). - Aditya Khanna, Dec 03 2025
MAPLE
# Uses EulerTransform from A358369.
t := EulerTransform(n -> padic[ordp](2*n, 2)):
seq(t(n), n = 0..41); # Peter Luschny, Nov 18 2022
MATHEMATICA
m = 42;
1/Product[QPochhammer[x^(2^k)], {k, 0, Log[2, m]//Ceiling}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Jan 14 2020, after Joerg Arndt *)
nmax = 60; CoefficientList[1/Series[Product[(1 - x^k)^(IntegerExponent[k, 2] + 1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 21 2026 *)
PROG
(PARI) N=66; x='x+O('x^N); /* that many terms */
gf=1/prod(e=0, ceil(log(N)/log(2)), eta(x^(2^e)));
Vec(gf) /* show terms */ /* Joerg Arndt, Jun 21 2011 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Mar 29 2004
STATUS
approved