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A156217
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^n*x^n/n ), a power series in x with integer coefficients.
5
1, 1, 5, 26, 634, 2273, 502568, 821149, 323391480, 1514316108, 360153555251, 440146271717, 19353735645240631, 19423959923754863, 1599762850473552085, 35664862808194240282, 45517618403498070780338, 45844669861151626268272
OFFSET
0,3
COMMENTS
Logarithmic derivative yields A217872.
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.
LINKS
FORMULA
a(n) = (1/n)*Sum_{k=1..n} sigma(k)^k * a(n-k) for n>0, with a(0)=1.
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 2273*x^5 + 502568*x^6 + ...
log(A(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 6^5*x^5/5 + 12^6*x^6/6 + ...
MAPLE
A156217 := proc(n) option remember ; if n = 0 then 1; else add( (numtheory[sigma](k))^k*procname(n-k), k=1..n)/n ; fi; end: seq(A156217(n), n=0..10) ; # R. J. Mathar, Apr 02 2009
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(m)^m*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^k*a(n-k)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 06 2009
STATUS
approved