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A158441 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1+x^n) /n ).

%I A158441 #39 May 05 2023 01:45:07
%S A158441 1,1,1,3,2,4,7,7,9,14,18,20,31,34,42,61,69,83,109,127,156,203,228,276,
%T A158441 347,404,477,591,683,801,990,1132,1323,1598,1837,2148,2560,2929,3405,
%U A158441 4018,4608,5319,6244,7124,8184,9569,10877,12465,14457,16412,18761,21633
%N A158441 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1+x^n) /n ).
%C A158441 Also the number of partitions of n in which each part occurs a triangle number (>=0) times. - _Seiichi Manyama_, May 11 2018
%H A158441 Alois P. Heinz, <a href="/A158441/b158441.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Seiichi Manyama)
%F A158441 Euler transform of A048272. [_Vladeta Jovovic_, Mar 28 2009]
%F A158441 G.f.: 1/prod(n>=1, P(x^n)^((-1)^(n-1)) ) where P(x) = prod(k>=1, 1-x^k ), see Pari code. [_Joerg Arndt_, Jun 24 2011]
%F A158441 G.f.: Product_{k>0} (Sum_{m>=0} x^(k*m*(m+1)/2)) = (1+x+x^3+x^6+...)*(1+x^2+x^6+x^12+...)*(1+x^3+x^9+x^18+...)*... . - _Seiichi Manyama_, May 11 2018
%F A158441 a(n) ~ (log(2))^(3/8) * exp(Pi*sqrt(2*log(2)*n/3)) / (2^(11/8) * 3^(3/8) * Pi^(1/4) * n^(7/8)). - _Vaclav Kotesovec_, Oct 08 2018
%e A158441 From _Seiichi Manyama_, Mar 11 2018: (Start)
%e A158441 n | Partitions of n in which each part occurs a triangle number (>=0) times.
%e A158441 --+-------------------------------------------------------------------------
%e A158441 1 | 1;
%e A158441 2 | 2;
%e A158441 3 | 3 = 2+1 = 1+1+1;
%e A158441 4 | 4 = 3+1;
%e A158441 5 | 5 = 4+1 = 3+2 = 2+1+1+1;
%e A158441 6 | 6 = 5+1 = 4+2 = 3+2+1 = 3+1+1+1 = 2+2+2 = 1+1+1+1+1+1;
%e A158441 7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 4+1+1+1 = 2+2+2+1; (End)
%p A158441 with(numtheory):
%p A158441 b:= proc(n) option remember; -add((-1)^d, d=divisors(n)) end:
%p A158441 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A158441       d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
%p A158441     end:
%p A158441 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 11 2018
%t A158441 nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 08 2018 *)
%o A158441 (PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, sigma(m)*x^m/(1+x^m+x*O(x^n))/m)), n))}
%o A158441 (PARI) N=99; x='x+O('x^N);
%o A158441 gf=1/prod(n=1,N,eta(x^n)^((-1)^(n-1)));
%o A158441 Vec(gf) /* _Joerg Arndt_, Jun 24 2011 */
%Y A158441 Cf. A006171, A000203, A295794, A320250.
%K A158441 nonn
%O A158441 0,4
%A A158441 _Paul D. Hanna_, Mar 28 2009