cp's OEIS Frontend

A300301 Number of ways to choose a partition, with odd parts, of each part of a partition of n into odd parts.

%I A300301 #8 Mar 05 2018 17:55:22
%S A300301 1,1,1,3,3,6,10,15,21,37,56,80,127,183,280,428,616,893,1367,1944,2846,
%T A300301 4223,6049,8691,12670,18128,25921,37529,53338,75738,108561,153460,
%U A300301 216762,308829,433893,612006,864990,1211097,1697020,2386016,3331037,4648229,6503314
%N A300301 Number of ways to choose a partition, with odd parts, of each part of a partition of n into odd parts.
%H A300301 Alois P. Heinz, <a href="/A300301/b300301.txt">Table of n, a(n) for n = 0..5000</a>
%F A300301 O.g.f.: Product_{n odd} 1/(1 - A000009(n)x^n).
%e A300301 The a(6) = 10 twice-partitions using odd partitions: (5)(1), (3)(3), (113)(1), (3)(111), (111)(3), (3)(1)(1)(1), (11111)(1), (111)(111), (111)(1)(1)(1), (1)(1)(1)(1)(1)(1).
%p A300301 with(numtheory):
%p A300301 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A300301      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
%p A300301     end:
%p A300301 g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A300301       g(n, i-2)+`if`(i>n, 0, b(i)*g(n-i, i)))
%p A300301     end:
%p A300301 a:= n-> g(n, n-1+irem(n,2)):
%p A300301 seq(a(n), n=0..50);  # _Alois P. Heinz_, Mar 05 2018
%t A300301 nn=50;
%t A300301 ser=Product[1/(1-PartitionsQ[n]x^n),{n,1,nn,2}];
%t A300301 Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]
%Y A300301 A300300(n) <= a(n) <= A279785(n)
%Y A300301 Cf. A000009, A063834, A078408, A089259, A270995, A271619, A279374, A279375, A279790, A294617, A300300.
%K A300301 nonn
%O A300301 0,4
%A A300301 _Gus Wiseman_, Mar 02 2018