A224439 -id:A224439 - cp's OEIS frontend

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

Paul D. Hanna, Feb 06 2009

nonn

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.

			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 + ...

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

Cf. A217872, A224439, A000041, A000203.

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

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

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

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

A224440 a(n) = sigma(n)^(n-1).

Original entry on oeis.org

1, 3, 16, 343, 1296, 248832, 262144, 170859375, 815730721, 198359290368, 61917364224, 8293509467471872, 56693912375296, 876488338465357824, 21035720123168587776, 23465261991844685929951, 121439531096594251776, 1117116121846700839825703079, 262144000000000000000000

Offset: 1

Paul D. Hanna, Apr 06 2013

nonn

Here sigma(n) = A000203(n) is the sum of the divisors of n.

			L.g.f.: L(x) = x + 3^1*x^2/2 + 4^2*x^3/3 + 7^3*x^4/4 + 6^4*x^5/5 + 12^5*x^6/6 +...
where exponentiation yields the g.f. of A224439:
exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 93*x^4 + 357*x^5 + 41927*x^6 +...

Seiichi Manyama, Table of n, a(n) for n = 1..335

Cf. A224439, A217872.

{a(n)=sigma(n)^(n-1)}
for(n=1, 20, print1(a(n), ", "))

Logarithmic derivative of A224439.

Offset corrected by Seiichi Manyama, Nov 10 2017

cp's OEIS Frontend

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

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A224440 a(n) = sigma(n)^(n-1).

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