A198299 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n).
1, 3, 4, 11, 6, 36, 8, 83, 49, 178, 12, 680, 14, 920, 714, 2707, 18, 7119, 20, 14166, 7844, 22564, 24, 94616, 3931, 106538, 88987, 306604, 30, 832606, 32, 1401715, 974736, 2228278, 150758, 9643703, 38, 9961532, 10363682, 28802278, 42, 78793604, 44, 123016344
Offset: 1
Keywords
Examples
L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 36*x^6/6 +...
such that, by definition:
L(x) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)) + (x^3/3)/((1-x^3)*(1-3*x^3)) + (x^4/4)/((1-x^4)*(1-2*x^4)*(1-4*x^4)) + (x^5/5)/((1-x^5)*(1-5*x^5)) + (x^6/6)/((1-x^6)*(1-2*x^6)*(1-3*x^6)*(1-6*x^6)) +...+ (x^n/n)/Product_{d|n} (1-d*x^n) +...
Also, we have the identity:
L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x
+ (1 + 3*x^2 + 7*x^4 + 15*x^6 + 31*x^8 +...)*x^2/2
+ (1 + 4*x^3 + 13*x^6 + 40*x^9 + 121*x^12 +...)*x^3/3
+ (1 + 7*x^4 + 35*x^8 + 155*x^12 + 651*x^16 +...)*x^4/4
+ (1 + 6*x^5 + 31*x^10 + 156*x^15 + 781*x^20 +...)*x^5/5
+ (1 + 12*x^6 + 97*x^12 + 672*x^18 + 4333*x^24 +...)*x^6/6 +...
+ exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k )*x^n/n +...
Exponentiation yields the g.f. of A198296:
exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 17*x^6 + 22*x^7 +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..500
Programs
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PARI
{a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sum(k=1, n\m, sigma(m, k)*x^(m*k)/k)+x*O(x^n))), n)} -
PARI
{a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d,-log(1-d*x^m+x*O(x^n))))), n)}
Formula
L.g.f.: Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n,k) * x^(n*k)/k ), where sigma(n,k) is the sum of the k-th powers of the divisors of n.
Comments