A156304 - OEIS

%I #14 Nov 02 2024 12:31:04

%S 1,1,8,21,77,199,661,1663,4852,12382,33289,82877,213026,518109,

%T 1279852,3053404,7312985,17093793,39952528,91661695,209709116,

%U 473095589,1062567288,2359804486,5214774263,11415904502,24860918943,53709881911

%N G.f.: A(x) = exp( Sum_{n>=1} sigma(n^3)*x^n/n ), a power series in x with integer coefficients.

%C Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

%C Euler transform of A160889. - _Vaclav Kotesovec_, Nov 01 2024

%H Seiichi Manyama, <a href="/A156304/b156304.txt">Table of n, a(n) for n = 0..5000</a>

%F a(n) = (1/n)*Sum_{k=1..n} sigma(k^3) * a(n-k) for n>0, with a(0)=1.

%F log(a(n)) ~ 4*Pi*c^(1/4)*n^(3/4) / (3^(5/4)*5^(1/4)), where c = A330595 = Product_{primes p} (1 + 1/p^2 + 1/p^3) = 1.74893299784324530303390699... - _Vaclav Kotesovec_, Nov 01 2024

%e G.f.: A(x) = 1 + x + 8*x^2 + 21*x^3 + 77*x^4 + 199*x^5 + 661*x^6 +...

%e log(A(x)) = x + 15*x^2/2 + 40*x^3/3 + 127*x^4/4 + 156*x^5/5 + 600*x^6/6 +...

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

%o (PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k^3)*a(n-k)))}

%Y Cf. A000203 (sigma), A000041 (partitions), A156303, A202993, A203557.

%Y Cf. A160889, A175926, A330595.

%K nonn,changed

%O 0,3

%A _Paul D. Hanna_, Feb 08 2009