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A156302 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^2*x^n/n ), a power series in x with integer coefficients. 5
1, 1, 5, 10, 30, 57, 152, 289, 676, 1304, 2809, 5335, 10961, 20487, 40329, 74476, 141914, 258094, 479638, 860025, 1563716, 2767982, 4940567, 8636563, 15173805, 26217392, 45416811, 77629455, 132800937, 224695510, 380079521, 637006921 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare with g.f. for 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

Table of n, a(n) for n=0..31.

FORMULA

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

Euler transform of A060648. [From Vladeta Jovovic, Feb 14 2009]

It appears that G.f.: A(x)=prod(n=1,infinity, E(x^n)^(-A001615(n))) where E(x) = prod(n=1,infinity,1-x^n). [From Joerg Arndt, Dec 30 2010]

G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k) * x^(n*k) / n ). [From Paul D. Hanna, Jan 23 2012]

EXAMPLE

G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 57*x^5 + 152*x^6 +...

log(A(x)) = x + 3^2*x^2/2 + 4^2*x^3/3 + 7^2*x^4/4 + 6^2*x^5/5 + 12^2*x^6/6 +...

Also log(A(x)) = (x + 3*x^2 + 4*x^3 + 7*x^4 +...+ sigma(k)*x^k +...)/1 +

(3*x^2 + 7*x^4 + 12*x^6 + 15*x^8 + 18*x^10 +...+ sigma(2*k)*x^(2*k) +...)/2 +

(4*x^3 + 12*x^6 + 13*x^9 + 28*x^12 + 24*x^15 +...+ sigma(3*k)*x^(3*k) +...)/3 +

(7*x^4 + 15*x^8 + 28*x^12 + 31*x^16 + 42*x^20 +...+ sigma(4*k)*x^(4*k) +...)/4 +

(6*x^5 + 18*x^10 + 24*x^15 + 42*x^20 + 31*x^25 +...+ sigma(5*k)*x^(5*k) +...)/5 +...

PROG

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

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

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

CROSSREFS

Cf. A000203 (sigma), A000041 (partitions).

Sequence in context: A069921 A053818 A133629 * A156234 A048010 A002571

Adjacent sequences:  A156299 A156300 A156301 * A156303 A156304 A156305

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 08 2009

STATUS

approved

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Last modified December 4 09:05 EST 2016. Contains 278749 sequences.