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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. 13
 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 (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 Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 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 A254128 Adjacent sequences:  A156300 A156301 A156302 * A156304 A156305 A156306 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 08 2009 STATUS approved

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Last modified May 27 17:53 EDT 2020. Contains 334664 sequences. (Running on oeis4.)