%I #52 Sep 07 2023 15:46:32
%S 1,1,2,5,7,15,25,43,64,120,186,288,463,695,1105,1728,2525,3741,5775,
%T 8244,12447,18302,26424,37827,54729,78330,111184,159538,225624,315415,
%U 444708,618666,858165,1199701,1646076,2288961,3150951,4303995,5870539,8032571,10881794,14749051,19992626
%N Expansion of Product_{m>=1} (1 + m*q^m).
%C Sum of products of terms in all partitions of n into distinct parts. - _Vladeta Jovovic_, Jan 19 2002
%C Number of partitions of n into distinct parts, when there are j sorts of part j. a(4) = 7: 4, 4', 4'', 4''', 31, 3'1, 3''1. - _Alois P. Heinz_, Aug 24 2015
%H Alois P. Heinz, <a href="/A022629/b022629.txt">Table of n, a(n) for n = 0..10000</a>
%H Vaclav Kotesovec, <a href="/A022629/a022629.jpg">Graph - The asymptotic ratio (1000000 terms)</a>
%F Conjecture: log(a(n)) ~ sqrt(n/2) * (log(2*n) - 2). - _Vaclav Kotesovec_, May 08 2018
%e The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding products are 6,5,8,6 and their sum is a(6) = 25.
%p b:= proc(n, i) option remember; local f, g;
%p if n=0 then [1, 1] elif i<1 then [0, 0]
%p else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i-1));
%p [f[1]+g[1], f[2]+g[2]*i]
%p fi
%p end:
%p a:= n-> b(n, n)[2]:
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Nov 02 2012
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, i*b(n-i, i-1))))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Aug 24 2015
%t nn=20;CoefficientList[Series[Product[1+i x^i,{i,1,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Nov 02 2012 *)
%t nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(j+1)*PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 28 2015 *)
%t (* More efficient program: 10000 terms, 4 minutes, 100000 terms, 6 hours *) nmax = 40; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j+1]] += k*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Jan 06 2016 *)
%o (PARI) N=66; q='q+O('q^N); Vec(prod(n=1,N, (1+n*q^n) )) \\ _Joerg Arndt_, Oct 06 2012
%o (Magma) Coefficients(&*[(1+m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 16 2018
%Y Cf. A000009, A006906, A022661, A022693, A265758, A266891, A285222, A304043.
%Y Cf. A092484, A265840, A265841, A265842, A322380, A322381.
%Y Column k=1 of A292189.
%K nonn
%O 0,3
%A _N. J. A. Sloane_