A318977 -id:A318977 - cp's OEIS frontend

A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij))^(1/(ij)).

1, 1, 2, 10, 34, 218, 1708, 12556, 97340, 1139932, 12602584, 142757624, 1983086488, 26745019000, 402951386576, 7181178238672, 115410887636752, 2039658743085584, 42354537803172640, 815690033731561888, 17593347085888752416, 416765224159172991136, 9379433694333768563392

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..446
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.

Cf. A000005, A107742, A168243, A280541, A288571, A318695, A318977.

seq(n!*coeff(series(mul((1+x^k)^(tau(k)/k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Product[Product[(1 + x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
nmax = 22; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, 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 01 2018 *)

E.g.f.: Product_{k>=1} (1 + x^k)^(tau(k)/k), where tau = number of divisors (A000005).

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*tau(d) ) * x^k/k).

A318695 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij))^(1/(ij)).

Original entry on oeis.org

1, 1, 4, 16, 106, 658, 6088, 51952, 592828, 6577948, 88213744, 1173121024, 18663391096, 289030343704, 5157010548064, 92428084599232, 1848308567352592, 37038307949425168, 822602470902709312, 18285742807660340992, 444405771941314880416, 10883864256927386369056, 286778106663948874858624

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

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.

Cf. A000005, A006171, A007425, A028342, A280540, A305127, A318696, A318977.

seq(n!*coeff(series(mul(1/(1-x^k)^(tau(k)/k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Product[Product[1/(1 - x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau(k)/k), where tau = number of divisors (A000005).

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} tau(d) ) * x^k/k).

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).

cp's OEIS Frontend

A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j))^(1/(i*j)).

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A318695 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(1/(i*j)).

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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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A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij))^(1/(ij)).

A318695 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(ij))^(1/(ij)).