A302830 - OEIS

%I #13 Apr 14 2018 03:45:38

%S 1,2,5,11,25,50,106,203,401,755,1427,2597,4804,8566,15352,27027,47551,

%T 82187,142445,243025,414919,700739,1181236,1972552,3293898,5450728,

%U 9008081,14791741,24244399,39494615,64266141,103979929,167991853,270190879,433773933,693518984

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

%C Partial sums of A006906.

%H Vaclav Kotesovec, <a href="/A302830/b302830.txt">Table of n, a(n) for n = 0..6000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F G.f.: (1/(1 - x))*exp(Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j).

%F From _Vaclav Kotesovec_, Apr 14 2018: (Start)

%F a(n) ~ c * 3^(n/3), where

%F c = 319343.48587983201292657132469068725642363369445... if mod(n,3)=0

%F c = 319343.34569378454521307030620964478962032866022... if mod(n,3)=1

%F c = 319343.21458897980925594955657564398036486423380... if mod(n,3)=2

%F (End)

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p        b(n, i-1)+add(b(n-i*j, i-1)*(i^j), j=1..n/i)))

%p     end:

%p a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Apr 13 2018

%t nmax = 35; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 35; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]

%Y Cf. A000041, A000070, A006906, A091360, A302831.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 13 2018