A158441 - OEIS

%I #39 May 05 2023 01:45:07

%S 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 347,404,477,591,683,801,990,1132,1323,1598,1837,2148,2560,2929,3405,

%U 4018,4608,5319,6244,7124,8184,9569,10877,12465,14457,16412,18761,21633

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

%C Also the number of partitions of n in which each part occurs a triangle number (>=0) times. - _Seiichi Manyama_, May 11 2018

%H 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 Euler transform of A048272. [_Vladeta Jovovic_, Mar 28 2009]

%F 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 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 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 From _Seiichi Manyama_, Mar 11 2018: (Start)

%e n | Partitions of n in which each part occurs a triangle number (>=0) times.

%e --+-------------------------------------------------------------------------

%e 1 | 1;

%e 2 | 2;

%e 3 | 3 = 2+1 = 1+1+1;

%e 4 | 4 = 3+1;

%e 5 | 5 = 4+1 = 3+2 = 2+1+1+1;

%e 6 | 6 = 5+1 = 4+2 = 3+2+1 = 3+1+1+1 = 2+2+2 = 1+1+1+1+1+1;

%e 7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 4+1+1+1 = 2+2+2+1; (End)

%p with(numtheory):

%p b:= proc(n) option remember; -add((-1)^d, d=divisors(n)) end:

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p       d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, May 11 2018

%t 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 (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 (PARI) N=99; x='x+O('x^N);

%o gf=1/prod(n=1,N,eta(x^n)^((-1)^(n-1)));

%o Vec(gf) /* _Joerg Arndt_, Jun 24 2011 */

%Y Cf. A006171, A000203, A295794, A320250.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Mar 28 2009