cp's OEIS Frontend

A279374 Number of ways to choose an odd partition of each part of an odd partition of 2n+1.

%I A279374 #12 Dec 12 2016 13:41:55
%S A279374 1,3,6,15,37,80,183,428,893,1944,4223,8691,18128,37529,75738,153460,
%T A279374 308829,612006,1211097,2386016,4648229,9042678,17528035,33645928,
%U A279374 64508161,123178953,233709589,442583046,834923483,1567271495,2935406996,5481361193,10191781534
%N A279374 Number of ways to choose an odd partition of each part of an odd partition of 2n+1.
%C A279374 An odd partition is an integer partition of an odd number with an odd number of parts, all of which are odd.
%H A279374 Alois P. Heinz, <a href="/A279374/b279374.txt">Table of n, a(n) for n = 0..4919</a>
%H A279374 Gus Wiseman, <a href="/A279374/a279374.png">"Twice-odd partitions of n=9."</a>
%e A279374 The a(3)=15 ways to choose an odd partition of each part of an odd partition of 7 are:
%e A279374 ((7)), ((511)), ((331)), ((31111)), ((1111111)), ((5)(1)(1)), ((311)(1)(1)),
%e A279374 ((11111)(1)(1)), ((3)(3)(1)), ((3)(111)(1)), ((111)(3)(1)), ((111)(111)(1)),
%e A279374 ((3)(1)(1)(1)(1)), ((111)(1)(1)(1)(1)), ((1)(1)(1)(1)(1)(1)(1)).
%p A279374 g:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A279374       [0, 2, 0, 1$4, 2, 0, 2, 1$4, 0, 2][1+irem(d, 16)],
%p A279374       d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A279374     end:
%p A279374 b:= proc(n, i, t) option remember;
%p A279374       `if`(n=0, t, `if`(i<1, 0, b(n, i-2, t)+
%p A279374       `if`(i>n, 0, b(n-i, i, 1-t)*g((i-1)/2))))
%p A279374     end:
%p A279374 a:= n-> b(2*n+1$2, 0):
%p A279374 seq(a(n), n=0..35);  # _Alois P. Heinz_, Dec 12 2016
%t A279374 nn=20;Table[SeriesCoefficient[Product[1/(1-PartitionsQ[k]x^k),{k,1,2n-1,2}],{x,0,2n-1}],{n,nn}]
%Y A279374 Cf. A000009 (strict partitions), A078408 (odd partitions), A063834, A271619, A279375.
%K A279374 nonn
%O A279374 0,2
%A A279374 _Gus Wiseman_, Dec 11 2016