A318413 -id:A318413 - cp's OEIS frontend

A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))^(ijk).

1, 1, 6, 15, 48, 108, 323, 716, 1868, 4217, 10137, 22311, 51477, 110817, 245260, 519918, 1114914, 2318557, 4854952, 9923533, 20335761, 40941170, 82365742, 163413699, 323589060, 633429923, 1236392498, 2390718266, 4606489839, 8805346615, 16768968317, 31713677061, 59747953446

Offset: 0

Ilya Gutkovskiy, Aug 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
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, A007425, A034718, A280473, A280541, A318413.

a:=series(mul(mul(mul((1+x^(i*j*k))^(i*j*k),k=1..55),j=1..55),i=1..55),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019

nmax = 32; CoefficientList[Series[Product[Product[Product[(1 + x^(i j k))^(i j k), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(k Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}]  x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
nmax = 32; A034718 = Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[A034718[[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] (* Vaclav Kotesovec, Aug 31 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^(k*tau_3(k)), where tau_3() = A007425.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^2 * Sum_{j|d} tau(j) ) * x^k/k), where tau() = A000005.
Conjecture: log(a(n)) ~ 3^(2/3) * Zeta(3)^(1/3) * log(n)^(2/3) * n^(2/3) / 2^(5/3). - Vaclav Kotesovec, Sep 02 2018

A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - ijk*x^(ijk)).

Original entry on oeis.org

1, 1, 7, 16, 64, 133, 465, 1008, 3023, 6695, 18206, 40175, 103229, 225470, 549873, 1194620, 2801742, 6015042, 13686306, 29063919, 64424496, 135362432, 293512852, 610061141, 1298516539, 2670738781, 5591712472, 11388116508, 23499720744, 47403692965, 96564236754

Offset: 0

Vaclav Kotesovec, Aug 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A006906, A318415.

Cf. A000041, A280540, A318413, A318482.

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

A318750 a(n) = Sum_{k=1..n} k * tau_3(k), where tau_3 is A007425.

Original entry on oeis.org

1, 7, 16, 40, 55, 109, 130, 210, 264, 354, 387, 603, 642, 768, 903, 1143, 1194, 1518, 1575, 1935, 2124, 2322, 2391, 3111, 3261, 3495, 3765, 4269, 4356, 5166, 5259, 5931, 6228, 6534, 6849, 8145, 8256, 8598, 8949, 10149, 10272, 11406, 11535, 12327, 13137, 13551

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Cf. A007425, A034718, A061201, A318413, A318414.

Cf. A001620, A082633.

Accumulate[Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 100}]]
(* Asymptotics: *) n^2 * (Log[n]^2 + (6*EulerGamma - 1)*Log[n] + 6*EulerGamma^2 - 3*EulerGamma - 6*StieltjesGamma[1] + 1/2) / 4 (* Vaclav Kotesovec, Sep 09 2018 *)

tau_3(k) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(k)[, 2]));
a(n) = sum(k = 1, n,  k * tau_3(k)); \\ Amiram Eldar, Jan 18 2025

a(n) = Sum_{k=1..n} A034718(k).

a(n) ~ n^2 * (log(n)^2 + (6*g-1)*log(n) + 6*g^2 - 3*g - 6*g1 + 1/2) / 4, where g is the Euler-Mascheroni constant A001620 and g1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Sep 09 2018

A318764 Expansion of Product_{i>=1, j>=1, k>=1} ((1 + x^(ijk))/(1 - x^(ijk)))^(ijk).

Original entry on oeis.org

1, 2, 14, 44, 182, 548, 1932, 5632, 17654, 49872, 145020, 395256, 1090044, 2876424, 7606024, 19503312, 49850790, 124543772, 309436980, 755268832, 1831194724, 4376807896, 10387118328, 24359228520, 56720659372, 130737105940, 299256890672, 678941040784

Offset: 0

Vaclav Kotesovec, Sep 03 2018

nonn

Convolution of A318413 and A318414.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A318413, A318414.
Cf. A156616, A318579.
Cf. A015128, A305050.

nmax = 40; CoefficientList[Series[Product[Product[Product[((1 + x^(i*j*k))/(1 - x^(i*j*k)))^(i*j*k), {i, 1, nmax/j/k}], {j, 1, nmax/k}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k*Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x]

Conjecture: log(a(n)) ~ (21*Zeta(3))^(1/3) * log(n)^(2/3) * n^(2/3) / 2^(5/3).

A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(1/(ijk)).

Original entry on oeis.org

1, 1, 5, 21, 165, 1077, 11457, 103905, 1345257, 15834825, 237535389, 3372509709, 59235634125, 979573962429, 19224990899865, 366788042231193, 8019002662543953, 171360055378885905, 4132946756763614133, 97947895990285022085, 2576516749059849502581, 67124117357620005459141

Offset: 0

Ilya Gutkovskiy, Sep 06 2018

nonn

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, A007425, A007426, A028342, A174465, A318413, A318695, A318967.

a:=series(mul(mul(mul(1/(1-x^(i*j*k))^(1/(i*j*k)),k=1..21),j=1..50),i=1..50),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Apr 02 2019

nmax = 21; CoefficientList[Series[Product[Product[Product[1/(1 - x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Sum[Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau_3(k)/k), where tau_3 = A007425.

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).

cp's OEIS Frontend

A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(i*j*k).

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A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - i*j*k*x^(i*j*k)).

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A318750 a(n) = Sum_{k=1..n} k * tau_3(k), where tau_3 is A007425.

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A318764 Expansion of Product_{i>=1, j>=1, k>=1} ((1 + x^(i*j*k))/(1 - x^(i*j*k)))^(i*j*k).

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A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(1/(i*j*k)).

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A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))^(ijk).

A318481 Expansion of Product_{i>=1, j>=1, k>=1} 1/(1 - ijk*x^(ijk)).

A318764 Expansion of Product_{i>=1, j>=1, k>=1} ((1 + x^(ijk))/(1 - x^(ijk)))^(ijk).

A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(ijk))^(1/(ijk)).