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A279785 Number of ways to choose a strict partition of each part of a strict partition of n.

%I A279785 #23 Sep 18 2019 17:45:17
%S A279785 1,1,1,3,4,7,11,18,28,47,71,108,166,252,382,587,869,1282,1938,2832,
%T A279785 4153,6148,8962,12965,18913,27301,39380,56747,81226,115907,166358,
%U A279785 236000,334647,475517,671806,947552,1335679,1875175,2630584,3687589,5150585,7183548
%N A279785 Number of ways to choose a strict partition of each part of a strict partition of n.
%C A279785 This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -A000009(n). - _Seiichi Manyama_, Nov 14 2018
%H A279785 Seiichi Manyama, <a href="/A279785/b279785.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Alois P. Heinz)
%H A279785 Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>
%F A279785 G.f.: Product_{k>0} (1 + A000009(k)*x^k). - _Seiichi Manyama_, Nov 14 2018
%e A279785 The a(6)=11 twice-partitions are:
%e A279785 ((6)), ((5)(1)), ((51)), ((4)(2)), ((42)), ((41)(1)),
%e A279785 ((3)(2)(1)), ((31)(2)), ((32)(1)), ((321)), ((21)(2)(1)).
%p A279785 with(numtheory):
%p A279785 g:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A279785       `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
%p A279785     end:
%p A279785 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
%p A279785       `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, g(i)*b(n-i, i-1))))
%p A279785     end:
%p A279785 a:= n-> b(n$2):
%p A279785 seq(a(n), n=0..70);  # _Alois P. Heinz_, Dec 20 2016
%t A279785 nn=20;CoefficientList[Series[Product[(1+PartitionsQ[k]x^k),{k,nn}],{x,0,nn}],x]
%t A279785 (* Second program: *)
%t A279785 g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n]; b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, g[i]*b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Feb 07 2017, after _Alois P. Heinz_ *)
%Y A279785 Cf. A000009, A063834, A270995, A279375, A327553, A327605.
%K A279785 nonn
%O A279785 0,4
%A A279785 _Gus Wiseman_, Dec 18 2016