A201825 - OEIS

%I #15 Mar 30 2012 18:37:33

%S 1,0,1,1,4,2,12,6,26,21,61,46,155,114,317,293,704,627,1539,1400,3149,

%T 3124,6469,6463,13341,13467,26271,27889,51970,55513,101961,110261,

%U 195586,217238,373391,418299,708816,800833,1323842,1521372,2461081,2846722,4543987

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

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

%H Paul D. Hanna, <a href="/A201825/b201825.txt">Table of n, a(n) for n = 0..500</a>

%F 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.

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

%e G.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 2*x^5 + 12*x^6 + 6*x^7 + 26*x^8 +...

%e where

%e 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 +...

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

%o (PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,(sigma(k)^2 - sigma(k,2))/2 * a(n-k)))}

%Y Cf. A119616.

%K nonn

%O 0,5

%A _Paul D. Hanna_, Dec 05 2011