OFFSET
0,3
COMMENTS
Compare e.g.f. to exp( Sum_{n>=1} sigma(n) * x^n/n ) = Product_{n>=1} 1/(1 - x^n).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 22*x^3/3! + 178*x^4/4! + 1570*x^5/5! + 18808*x^6/6! + 230980*x^7/7! + 3505468*x^8/8! + 57144700*x^9/9! + 1068295600*x^10/10! + 20546428360*x^11/11! + 461887088728*x^12/12! + 10502814172696*x^13/13! + 264754450444576*x^14/14! + 7060121052388720*x^15/15! + 204307337026984720*x^16/16! +...
such that
log(A(x)) = 1*x + 1*3*x^2/2! + 1*3*4*x^3/3! + 1*3*4*7*x^4/4! + 1*3*4*7*6*x^5/5! + 1*3*4*7*6*12*x^6/6! + 1*3*4*7*6*12*8*x^7/7! + 1*3*4*7*6*12*8*15*x^8/8! + 1*3*4*7*6*12*8*15*13*x^9/9! + 1*3*4*7*6*12*8*15*13*18*x^10/10! +...+ (Product_{k=1..n} sigma(k))*x^n/n! +...
explicitly,
log(A(x)) = x + 3*x^2/2! + 12*x^3/3! + 84*x^4/4! + 504*x^5/5! + 6048*x^6/6! + 48384*x^7/7! + 725760*x^8/8! + 9434880*x^9/9! + 169827840*x^10/10! + 2037934080*x^11/11! + 57062154240*x^12/12! +...+ A066780(n)*x^n/n! +...
PROG
(PARI) {a(n) = n!*polcoeff( exp( sum(m=1, n+1, prod(k=1, m, sigma(k)) * x^m/m!) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 28 2017
STATUS
approved