cp's OEIS Frontend

A300300 Number of ways to choose a multiset of strict partitions, or odd partitions, of odd numbers, whose weights sum to n.

%I A300300 #10 Mar 05 2018 15:17:50
%S A300300 1,1,1,3,3,6,9,14,20,32,48,69,105,150,225,322,472,669,977,1379,1980,
%T A300300 2802,3977,5602,7892,11083,15494,21688,30147,42007,58143,80665,111199,
%U A300300 153640,211080,290408,397817,545171,744645,1016826,1385124,1885022,2561111,3474730
%N A300300 Number of ways to choose a multiset of strict partitions, or odd partitions, of odd numbers, whose weights sum to n.
%H A300300 Alois P. Heinz, <a href="/A300300/b300300.txt">Table of n, a(n) for n = 0..10000</a>
%F A300300 Euler transform of {Q(1), 0, Q(3), 0, Q(5), 0, ...} where Q = A000009.
%e A300300 The a(6) = 9 multiset partitions using odd-weight strict partitions: (5)(1), (14)(1), (3)(3), (32)(1), (3)(21), (3)(1)(1)(1), (21)(21), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1).
%e A300300 The a(6) = 9 multiset partitions using odd partitions: (5)(1), (3)(3), (311)(1), (3)(111), (3)(1)(1)(1), (11111)(1), (111)(111), (111)(1)(1)(1), (1)(1)(1)(1)(1)(1).
%p A300300 with(numtheory):
%p A300300 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A300300       `if`(d::odd, d, 0), d=divisors(j)), j=1..n)/n)
%p A300300     end:
%p A300300 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p A300300       `if`(d::odd, b(d)*d, 0), d=divisors(j)), j=1..n)/n)
%p A300300     end:
%p A300300 seq(a(n), n=0..45);  # _Alois P. Heinz_, Mar 02 2018
%t A300300 nn=50;
%t A300300 ser=Product[1/(1-x^n)^PartitionsQ[n],{n,1,nn,2}];
%t A300300 Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]
%Y A300300 Cf. A000009, A063834, A078408, A089259, A270995, A271619, A279374, A279375, A279785, A279790, A294617, A300301.
%K A300300 nonn
%O A300300 0,4
%A A300300 _Gus Wiseman_, Mar 02 2018