A180608 O.g.f.: exp( Sum_{n>=1} A067692(n)*x^n/n ), where A067692(n) = [sigma(n)^2 + sigma(n,2)]/2.

1, 1, 4, 8, 21, 39, 93, 171, 364, 675, 1338, 2433, 4641, 8282, 15222, 26811, 47920, 83046, 145288, 248164, 425970, 718303, 1213106, 2020540, 3365352, 5541996, 9115640, 14856657, 24164430, 39002462, 62800603, 100454208, 160257140

Offset: 0

Author

Paul D. Hanna, Oct 10 2010

Keywords

• nonn

Comments

sigma(n) = A000203(n), sum of divisors of n;

sigma(n,2) = A001157(n), sum of squares of divisors of n.

Examples

			O.g.f.: A(x) = 1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 93*x^6 +...
log(A(x)) = x + 7*x^2/2 + 13*x^3/3 + 35*x^4/4 + 31*x^5/5 + 97*x^6/6 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A067692, A000203, A001157, A072861, A156302.

Programs

Mathematica

nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[(DivisorSigma[1, k]^2 + DivisorSigma[2, k])/2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 28 2024 *)

PARI

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

Formula

log(a(n)) ~ 3*(7*zeta(3))^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Oct 29 2024