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A156303
G.f.: A(x) = exp( Sum_{n>=1} sigma(n^2)*x^n/n ), a power series in x with integer coefficients.
14
1, 1, 4, 8, 20, 38, 88, 162, 336, 624, 1211, 2195, 4109, 7295, 13190, 23072, 40618, 69838, 120486, 204006, 345595, 577387, 962961, 1588483, 2613930, 4262138, 6928799, 11179251, 17976330, 28720552, 45729595, 72401921, 114239033
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
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 Dedekind psi function, cf. A001615. - Vladeta Jovovic, Feb 12 2009
a(n) ~ exp(3^(4/3) * (5*Zeta(3))^(1/3) * n^(2/3) / (2*Pi)^(2/3) - Pi^(2/3) * n^(1/3) / (2^(4/3) * (15*Zeta(3))^(1/3)) + 1/12 - Pi^2 / (720 * Zeta(3))) * (5*Zeta(3))^(7/36) / (A * 2^(2/9) * 3^(11/36) * Pi^(29/36) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 24 2018
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 8*x^3 + 20*x^4 + 38*x^5 + 88*x^6 +...
log(A(x)) = x + 7*x^2/2 + 13*x^3/3 + 31*x^4/4 + 31*x^5/5 + 127*x^6/6 +...
MATHEMATICA
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*Sum[MoebiusMu[d]^2 / d, {d, Divisors @ k}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sigma(m^2)*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k^2)*a(n-k)))}
CROSSREFS
Cf. A000203 (sigma), A000041 (partitions), A001615, A301594.
Sequence in context: A097940 A032280 A300158 * A301138 A008136 A357060
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 08 2009
STATUS
approved