A201825 G.f.: exp( Sum_{n>=1} A119616(n)*x^n/n ) where A119616(n) = (sigma(n)^2 - sigma(n,2))/2.

1, 0, 1, 1, 4, 2, 12, 6, 26, 21, 61, 46, 155, 114, 317, 293, 704, 627, 1539, 1400, 3149, 3124, 6469, 6463, 13341, 13467, 26271, 27889, 51970, 55513, 101961, 110261, 195586, 217238, 373391, 418299, 708816, 800833, 1323842, 1521372, 2461081, 2846722, 4543987

Offset

0,5

Comments

Here sigma(n) = A000203(n), the sum of divisors of n; sigma(n,2) = A001157(n), the sum of squares of divisors of n.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

a(n) = (1/n)*Sum_{k=1..n} (sigma(k)^2 - sigma(k,2))/2 * a(n-k) for n>0 with a(0)=1.

Logarithmic derivative yields A119616, the second elementary symmetric function of divisors of n.

Example

G.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 2*x^5 + 12*x^6 + 6*x^7 + 26*x^8 +...
where
log(A(x)) = 2*x^2/2 + 3*x^3/3 + 14*x^4/4 + 5*x^5/5 + 47*x^6/6 + 7*x^7/7 + 70*x^8/8 + 39*x^9/9 + 97*x^10/10 + 11*x^11/11 +...+ A119616(n)*x^n/n +...

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(m)^2-sigma(m, 2))/2*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, (sigma(k)^2 - sigma(k, 2))/2 * a(n-k)))}

Crossrefs

Cf. A119616.

Sequence in context: A260434 A243344 A293603 * A104007 A191441 A152664

Adjacent sequences: A201822 A201823 A201824 * A201826 A201827 A201828

Keyword

nonn

Author

Paul D. Hanna, Dec 05 2011

Status

approved