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A317912 Expansion of Product_{k>=2} 1/(1 - k*x^k).

%I A317912 #10 Dec 08 2020 11:40:33
%S A317912 1,0,2,3,8,11,31,41,101,156,318,498,1037,1555,3024,4889,8849,14112,
%T A317912 25622,40322,71314,113926,194677,310819,530030,835484,1400523,2226307,
%U A317912 3668998,5797558,9521310,14942262,24298136,38187102,61384028,96161997,154078991,239891926,381723396
%N A317912 Expansion of Product_{k>=2} 1/(1 - k*x^k).
%C A317912 First differences of A006906.
%C A317912 Sum of products of terms in all partitions of n into parts >= 2.
%H A317912 Seiichi Manyama, <a href="/A317912/b317912.txt">Table of n, a(n) for n = 0..5000</a>
%H A317912 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A317912 G.f.: exp(Sum_{j>=1} Sum_{k>=2} k^j*x^(j*k)/j).
%e A317912 a(6) = 31 because we have [6], [4, 2], [3, 3], [2, 2, 2] and 6 + 4*2 + 3*3 + 2*2*2 = 31.
%p A317912 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A317912       b(n, i-1)+ i*b(n-i, min(n-i, i)))
%p A317912     end:
%p A317912 a:= n-> b(n$2) -`if`(n=0, 0, b(n-1$2)):
%p A317912 seq(a(n), n=0..40);  # _Alois P. Heinz_, Aug 10 2018
%t A317912 nmax = 38; CoefficientList[Series[Product[1/(1 - k x^k), {k, 2, nmax}], {x, 0, nmax}], x]
%t A317912 nmax = 38; CoefficientList[Series[Exp[Sum[Sum[k^j x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
%t A317912 b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d^(k/d + 1), {d, Divisors[k]}] b[n - k], {k, 1, n}]/n]; Differences[Table[b[n], {n, -1, 38}]]
%Y A317912 Cf. A002865, A006906, A191659, A302830.
%K A317912 nonn
%O A317912 0,3
%A A317912 _Ilya Gutkovskiy_, Aug 10 2018