A158441 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1+x^n) /n ).
1, 1, 1, 3, 2, 4, 7, 7, 9, 14, 18, 20, 31, 34, 42, 61, 69, 83, 109, 127, 156, 203, 228, 276, 347, 404, 477, 591, 683, 801, 990, 1132, 1323, 1598, 1837, 2148, 2560, 2929, 3405, 4018, 4608, 5319, 6244, 7124, 8184, 9569, 10877, 12465, 14457, 16412, 18761, 21633
Offset: 0
Keywords
Examples
From _Seiichi Manyama_, Mar 11 2018: (Start) n | Partitions of n in which each part occurs a triangle number (>=0) times. --+------------------------------------------------------------------------- 1 | 1; 2 | 2; 3 | 3 = 2+1 = 1+1+1; 4 | 4 = 3+1; 5 | 5 = 4+1 = 3+2 = 2+1+1+1; 6 | 6 = 5+1 = 4+2 = 3+2+1 = 3+1+1+1 = 2+2+2 = 1+1+1+1+1+1; 7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 4+1+1+1 = 2+2+2+1; (End)
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)
Programs
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Maple
with(numtheory): b:= proc(n) option remember; -add((-1)^d, d=divisors(n)) end: a:= proc(n) option remember; `if`(n=0, 1, add(add( d*b(d), d=divisors(j))*a(n-j), j=1..n)/n) end: seq(a(n), n=0..60); # Alois P. Heinz, May 11 2018 -
Mathematica
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 *) -
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))} -
PARI
N=99; x='x+O('x^N); gf=1/prod(n=1,N,eta(x^n)^((-1)^(n-1))); Vec(gf) /* Joerg Arndt, Jun 24 2011 */
Formula
Euler transform of A048272. [Vladeta Jovovic, Mar 28 2009]
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]
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
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
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