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A317913 Expansion of Product_{k>=2} (1 + k*x^k).

%I A317913 #13 Feb 12 2020 14:16:58
%S A317913 1,0,2,3,4,11,14,29,35,85,101,187,276,419,686,1042,1483,2258,3517,
%T A317913 4727,7720,10582,15842,21985,32744,45586,65598,93940,131684,183731,
%U A317913 260977,357689,500476,699225,946851,1342110,1808841,2495154,3375385,4657186,6224608,8524443,11468183,15428030
%N A317913 Expansion of Product_{k>=2} (1 + k*x^k).
%C A317913 Sum of products of terms in all partitions of n into distinct parts >= 2.
%H A317913 Robert Israel, <a href="/A317913/b317913.txt">Table of n, a(n) for n = 0..10000</a>
%H A317913 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A317913 G.f.: exp(Sum_{j>=1} Sum_{k>=2} (-1)^(j+1)*k^j*x^(j*k)/j).
%F A317913 a(n) + a(n-1) = A022629(n). - _Vaclav Kotesovec_, Aug 21 2018
%e A317913 a(7) = 29 because we have [7], [5, 2], [4, 3] and 7 + 5*2 + 4*3 = 29.
%p A317913 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A317913       b(n, i-1)+ i*b(n-i, min(n-i, i-1)))
%p A317913     end:
%p A317913 a:= n-> b(n$2) -`if`(n=0, 0, b(n-1$2)):
%p A317913 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 10 2018
%t A317913 nmax = 43; CoefficientList[Series[Product[(1 + k x^k), {k, 2, nmax}], {x, 0, nmax}], x]
%t A317913 nmax = 43; CoefficientList[Series[Exp[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
%Y A317913 Cf. A022629, A025147, A298598, A317912.
%K A317913 nonn
%O A317913 0,3
%A A317913 _Ilya Gutkovskiy_, Aug 10 2018