A318814 -id:A318814 - cp's OEIS frontend

A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

1, 1, 5, 23, 179, 1279, 13699, 135085, 1764377, 22527521, 344625461, 5283739471, 94562354875, 1685808248383, 33947023942259, 694786150879829, 15613612524749489, 357353282848083265, 8880505496901812197, 224851013929747732231, 6106205671049245677251, 169523515381173773551871

Offset: 0

Ilya Gutkovskiy, May 26 2018

nonn

a(n)/n! is the Euler transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Vaclav Kotesovec, Table of n, a(n) for n = 0..438
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

Cf. A000203, A007429, A017665, A017666, A028342, A053529, A061256, A072169, A318769, A318814.

with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(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

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[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[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]*(-1)^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}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018

A318975 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^phi(k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 4, 10, 20, 42, 80, 154, 288, 522, 940, 1658, 2892, 4970, 8456, 14218, 23696, 39122, 64044, 104042, 167732, 268602, 427248, 675482, 1061632, 1659298, 2579676, 3990418, 6142892, 9412906, 14360136, 21814698, 33004704, 49739426, 74677924, 111713658

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A299069 and A061255.

			a(n) ~ exp(3^(4/3) * (7*Zeta(3))^(1/3) * n^(2/3) / (2*Pi^(2/3)) - 1/6) * A^2 * (7*Zeta(3))^(1/9) / (sqrt(2) * 3^(7/18) * Pi^(8/9) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A061255, A299069, A301554, A301555.

Cf. A318976, A318814, A318977.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^EulerPhi[k], {k, 1, nmax}], {x, 0, nmax}], x]

A318976 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(phi(k)/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 6, 32, 196, 1512, 13384, 135872, 1545744, 19441952, 268386784, 4018603008, 65021744704, 1127284876928, 20880206388864, 410781080941568, 8561002328678656, 188224613741879808, 4355496092560324096, 105752112730661347328, 2688539359466319184896

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A088009 and A000262.

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A000262, A088009, A318814, A318977.

Cf. A318975, A301555, A301554.

nmax = 20; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(EulerPhi[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[E^(x*(2 + x)/(1 - x^2)), {x, 0, nmax}], x] * Range[0, nmax]!

E.g.f.: exp(x*(2 + x)/(1 - x^2)).

a(n) ~ 2^(-3/4) * 3^(1/4) * exp(sqrt(6*n) - n - 1/2) * n^(n - 1/4).

A318977 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(tau(k)/k), where tau is A000005.

Original entry on oeis.org

1, 2, 8, 44, 292, 2296, 21472, 221168, 2554544, 32617952, 452957056, 6788855872, 110098330048, 1900192498304, 34971968379392, 683452436531456, 14097619892177152, 306168410773570048, 6998327049216231424, 167369475021548506112, 4187842602663179396096

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A318696 and A318695.

Vaclav Kotesovec, Table of n, a(n) for n = 0..442

Cf. A318695, A318696, A318814, A318976.

Cf. A318975, A301555, A301554.

nmax = 20; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

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.

Original entry on oeis.org

1, 1, 3, 17, 83, 639, 5749, 53227, 561273, 7216577, 94292531, 1352253561, 21657812923, 359338829407, 6460367397093, 126124578755939, 2527688612931569, 54137820027005697, 1236730462664172643, 29137619131277727457, 725282418459957414051, 18981526480933601454911

Offset: 0

Ilya Gutkovskiy, Sep 03 2018

nonn

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, ...].

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
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

Cf. A000203, A017665, A017666, A168243, A192065, A288417, A305127, A318696, A318814.

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

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 *)

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

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A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

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A318975 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^phi(k), where phi is the Euler totient function A000010.

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A318976 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(phi(k)/k), where phi is the Euler totient function A000010.

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A318977 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(tau(k)/k), where tau is A000005.

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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.

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