A156217 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^n*x^n/n ), a power series in x with integer coefficients.

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

Seiichi Manyama, Table of n, a(n) for n = 0..335

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

Cf. A217872, A224439, A000041, A000203.

Sequence in context: A324419 A090644 A196951 * A156216 A320987 A015490

Adjacent sequences: A156214 A156215 A156216 * A156218 A156219 A156220

Keyword

nonn

Author

Paul D. Hanna, Feb 06 2009

Status

approved