A192065 -id:A192065 - cp's OEIS frontend

A321042 a(n) = [x^n] Product_{k>=1} (1 + x^k)^sigma_n(k).

1, 1, 5, 37, 491, 12763, 690756, 70250881, 13805853214, 5567873958982, 4386114219458332, 6711687353310594027, 21048327399504558833175, 131214860796100022696745520, 1603892616451767287785208156624, 40296605442098101265893075903063822, 2031406440758379976992019043333960734724

Offset: 0

Ilya Gutkovskiy, Oct 26 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A107742, A192065, A288414, A288415, A301548, A301549, A301550, A301551, A301552, A301553, A319647.

Table[SeriesCoefficient[Product[(1 + x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[Product[(1 + x^(i j))^(j^n), {j, 1, n}], {i, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^k/(k (1 - x^(2 k))), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

a(n) = [x^n] Product_{i>=1, j>=1} (1 + x^(i*j))^(j^n).

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^k/(k*(1 - x^(2*k)))).

A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

Original entry on oeis.org

1, 5, 13, 21, 31, 65, 57, 85, 130, 155, 133, 273, 183, 285, 403, 341, 307, 650, 381, 651, 741, 665, 553, 1105, 806, 915, 1210, 1197, 871, 2015, 993, 1365, 1729, 1535, 1767, 2730, 1407, 1905, 2379, 2635, 1723, 3705, 1893, 2793, 4030, 2765, 2257, 4433, 2850, 4030

Offset: 1

Seiichi Manyama, Jun 09 2017

Multiplicative because this sequence is the Dirichlet convolution of A000290 and A000593 which are both multiplicative. - Andrew Howroyd, Jul 27 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000290, A001001, A192065, A183699.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), this sequence (k=2), A288419 (k=3), A288420 (k=4).

a[n_] := DivisorSum[n, Function[d, d^2*DivisorSum[n/d, If[OddQ[#], #, 0]&]] ];
Array[a, 50] (* Jean-François Alcover, Jul 03 2017 *)
f[p_, e_] := (p^(e + 1) - 1)*(p^(e + 2) - 1)/((p - 1)*(p^2 - 1)); f[2, e_] := (4^(e + 1) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = sumdiv(n, d, d^2*sigma((n/d)>>valuation(n/d, 2))); \\ Michel Marcus, Jul 03 2017; corrected Jun 12 2022

L.g.f.: log(Product_{k>=1} (1 + x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jun 19 2018
From Amiram Eldar, Nov 13 2022: (Start)
a(n) = A001001(n) for odd n.
Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(e+1)-1)*(p^(e+2)-1)/((p-1)*(p^2-1)) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)*zeta(3)/4 = A183699 / 4 = 0.494326... . (End)

A288421 Expansion of Product_{k>=1} 1/(1 + x^k)^sigma(k).

Original entry on oeis.org

1, -1, -2, -2, 1, 5, 4, 10, 6, -5, -20, -27, -37, -32, -18, 23, 82, 128, 190, 185, 143, 43, -160, -424, -662, -968, -1058, -971, -571, 238, 1326, 2748, 4195, 5301, 5930, 5473, 3353, 55, -5346, -12106, -19421, -26603, -31950, -33248, -29344, -17469, 2343, 30966

Offset: 0

Seiichi Manyama, Jun 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A192065, A288418.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), this sequence (m=1), A288422 (m=2), A288423 (m=3).

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(1,k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

nmax = 50; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1+x^k)^sigma(k))) \\ G. C. Greubel, Oct 29 2018

Convolution inverse of A192065.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

A301799 Expansion of Product_{k>=1} 1/(1 - x^k)^A000593(k).

Original entry on oeis.org

1, 1, 2, 6, 8, 18, 34, 56, 98, 175, 290, 479, 809, 1293, 2096, 3382, 5324, 8378, 13140, 20319, 31328, 48098, 73096, 110763, 167100, 250365, 373670, 555613, 821604, 1210709, 1777718, 2598584, 3786132, 5498169, 7954764, 11473798, 16499790, 23650735, 33806012

Offset: 0

Vaclav Kotesovec, Mar 26 2018

nonn

Euler transform of A000593.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)

Cf. A000593, A002131, A192065, A301800.

nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, -(-1)^# k / # &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

a(n) ~ exp((3*Pi)^(2/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + 1/24) * Zeta(3)^(13/72) / (sqrt(A) * 2^(23/36) * 3^(49/72) * Pi^(13/72) * n^(49/72)), where A is the Glaisher-Kinkelin constant A074962.

G.f.: exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 + x^k))). - Ilya Gutkovskiy, Oct 26 2018

A301553 Expansion of Product_{k>=1} (1 + x^k)^(sigma_9(k)).

Original entry on oeis.org

1, 1, 513, 20197, 413669, 12445003, 372981573, 9158438541, 223776496101, 5567873958982, 132009631562091, 3018411978731059, 68171158091244082, 1512439928316217508, 32796174722883608382, 698503712498547606328, 14656105328324700415778, 302787437988353941515934

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..995

Cf. A107742 (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

Cf. A013957, A301547.

nmax = 30; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[9, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(11 * Pi^(10/11) * (31*Zeta(11))^(1/11) * n^(10/11) / (2^(13/11) * 5^(10/11))) * (155*Zeta(11)/Pi)^(1/22) / (2^(155/264) * sqrt(11) * n^(6/11)).

G.f.: exp(Sum_{k>=1} sigma_10(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 26 2018

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

A327063 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^j).

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 8, 11, 15, 24, 34, 43, 63, 87, 115, 159, 217, 279, 380, 505, 657, 868, 1139, 1458, 1913, 2482, 3162, 4069, 5232, 6628, 8469, 10755, 13544, 17127, 21634, 27061, 33988, 42557, 52985, 66069, 82289, 101862, 126281, 156275, 192655, 237530, 292502

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327064, A327065, A327066.

nmax = 50; CoefficientList[Series[Product[Product[(1+x^(k*j))^j, {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]

A327064 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^k).

Original entry on oeis.org

1, 1, 2, 5, 10, 18, 35, 62, 110, 197, 339, 573, 975, 1621, 2674, 4385, 7108, 11422, 18277, 28976, 45648, 71531, 111372, 172416, 265695, 407210, 621143, 943392, 1426414, 2147672, 3221271, 4812534, 7163440, 10625651, 15706871, 23141148, 33987287, 49762235

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327063, A327065, A327067.

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

A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(kj))^(kj)).

Original entry on oeis.org

1, 1, 2, 5, 12, 20, 42, 75, 141, 259, 466, 799, 1427, 2443, 4169, 7049, 11863, 19605, 32518, 53184, 86579, 140018, 225380, 359739, 572864, 905903, 1426270, 2234952, 3488313, 5416403, 8383226, 12917257, 19831763, 30334937, 46245977, 70242043, 106371686

Offset: 0

Vaclav Kotesovec, Aug 19 2019

nonn

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

Cf. A028377, A192065, A211856, A294842, A280541, A327063, A327064, A327068.

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

A301798 Expansion of Product_{k>=1} (1 + x^k)^A002131(k).

Original entry on oeis.org

1, 1, 2, 6, 9, 19, 36, 62, 110, 197, 332, 559, 947, 1548, 2538, 4133, 6610, 10536, 16710, 26191, 40879, 63465, 97732, 149852, 228658, 346788, 523694, 787503, 1178325, 1756294, 2607686, 3855676, 5680851, 8341007, 12202794, 17795283, 25869297, 37487313

Offset: 0

Vaclav Kotesovec, Mar 26 2018

nonn

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

Cf. A000203, A002131, A192065.

nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^j * Sum[DivisorSum[k, # / GCD[#, 2] &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

a(n) ~ exp(3^(4/3) * Pi^(2/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(7/3) - Pi^(4/3) * n^(1/3) / (2^(8/3) * 3^(4/3) * Zeta(3)^(1/3)) - Pi^2 / (2592 * Zeta(3))) * Zeta(3)^(1/6) / (2^(7/6) * 3^(1/3) * Pi^(1/6) * n^(2/3)).

cp's OEIS Frontend

A321042 a(n) = [x^n] Product_{k>=1} (1 + x^k)^sigma_n(k).

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A288418 a(n) = Sum_{d|n} d^2*A000593(n/d).

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A288421 Expansion of Product_{k>=1} 1/(1 + x^k)^sigma(k).

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A301799 Expansion of Product_{k>=1} 1/(1 - x^k)^A000593(k).

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A301553 Expansion of Product_{k>=1} (1 + x^k)^(sigma_9(k)).

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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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A327063 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^j).

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A327064 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^k).

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A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))^(k*j)).

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A327065 Expansion of Product_{k>=1} (Product_{j=1..k} (1 + x^(kj))^(kj)).