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A318769
Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.
4
1, 1, 3, 17, 83, 639, 5749, 53227, 561273, 7216577, 94292531, 1352253561, 21657812923, 359338829407, 6460367397093, 126124578755939, 2527688612931569, 54137820027005697, 1236730462664172643, 29137619131277727457, 725282418459957414051, 18981526480933601454911
OFFSET
0,3
COMMENTS
a(n)/n! is the weigh transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].
LINKS
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms
FORMULA
E.g.f.: Product_{k>=1} (1 + x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n/2) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018
a(n)/n! ~ c * exp(sqrt(n/2)*Pi^2/3) / n^(3/4 + log(2)/4), where c = 0.15653645678497413538057076667218805302154965061194080137... - Vaclav Kotesovec, Sep 05 2018
MAPLE
with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(-(-1)^(j/d)*sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018
MATHEMATICA
nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 03 2018
STATUS
approved