A182819 - OEIS

%I #24 Jan 26 2017 02:45:47

%S 1,4,14,39,101,238,533,1131,2314,4566,8763,16376,29939,53612,94302,

%T 163112,277953,467064,774943,1270528,2060331,3306771,5256579,8280649,

%U 12934125,20040761,30817437,47048638,71339593,107469716,160898163

%N G.f.: exp( Sum_{n>=1} sigma(3n)*x^n/n ).

%C sigma(3n) = A000203(3n), the sum of divisors of 3n (A144613).

%C Compare g.f. to P(x), the g.f. of partition numbers (A000041): P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ).

%C In general, if r>0 and g.f. = Product_{k>=1} (1 - x^(r*k))/(1 - x^k)^(r+1) then a(n) ~ (r+1-1/r)^((r+1)/4) * exp(Pi*sqrt(2*(r+1-1/r)*n/3)) / (sqrt(r) * 2^((3*r+5)/4) * 3^((r+1)/4) * n^((r+3)/4)). - _Vaclav Kotesovec_, Nov 28 2016

%H Seiichi Manyama, <a href="/A182819/b182819.txt">Table of n, a(n) for n = 0..1000</a>

%F Generating function A(x) = E(x^3)/E(x)^4 where E(x) = Product_{n>=1} (1-x^n). [_Joerg Arndt_, Dec 05 2010]

%F a(n) ~ 11*exp(sqrt(22*n)*Pi/3) / (72*sqrt(6)*n^(3/2)). - _Vaclav Kotesovec_, Nov 26 2016

%F From _Peter Bala_, Jan 24 2016: (Start)

%F A(x^3) = P(x)*P(w*x)*P(w^2*x), where P(x) = 1/Product_{n>=1} (1 - x^n) is the g.f. for the partition function p(n) = A000041(n), and where w = exp(2*Pi*i/3) is a primitive cube root of unity.

%F a(n) = Sum_{j = 0..3*n} ( Sum_{k = 0..3*n-j} w^(j+2*k)*p(k)*p(j) *p(3*n-j-k) ). (End)

%e G.f.: A(x) = 1 + 4*x + 14*x^2 + 39*x^3 + 101*x^4 + 238*x^5 +...

%e log(A(x)) = 4*x + 12*x^2/2 + 13*x^3/3 + 28*x^4/4 + 24*x^5/5 + 39*x^6/6 + 32*x^7/7 + 60*x^8/8 +...+ sigma(3n)*x^n/n +...

%p w := exp(2*Pi*sqrt(-1)*(1/3)):

%p with(combinat):

%p seq(simplify(add(add(w^(j+2*k)*numbpart(j)*numbpart(k)*numbpart(3*n-j-k), k = 0..3*n-j), j = 0..3*n)), n = 0..30);

%p # _Peter Bala_, Jan 24 2017

%t nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 26 2016 *)

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sigma(3*m)*x^m/m)+x*O(x^n)),n)}

%o (PARI) default(seriesprecision,66); Vec(eta(x^3)/eta(x)^4)\\ _Joerg Arndt_, Dec 06 2010

%Y Cf. A144613, A000203, A000041; variants: A182818, A182820, A182821.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Dec 05 2010