A318784 -id:A318784 - cp's OEIS frontend

A318783 Expansion of Product_{k>=1} 1/(1 - x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

1, 0, 1, 1, 3, 2, 7, 5, 14, 13, 27, 26, 57, 53, 102, 110, 192, 204, 353, 381, 626, 704, 1094, 1246, 1920, 2185, 3252, 3800, 5503, 6440, 9213, 10827, 15194, 18035, 24836, 29579, 40369, 48103, 64758, 77635, 103279, 123882, 163506, 196286, 256688, 308836, 400329, 481847, 620832

Offset: 0

Ilya Gutkovskiy, Sep 03 2018

nonn

Convolution of A010815 and A006171.

Euler transform of A032741.

N. J. A. Sloane, Transforms

Cf. A000005, A000203, A006171, A010815, A032741, A318784.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      (tau(d)-1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 03 2018

nmax = 48; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[0, k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^(2 k)/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (DivisorSigma[0, d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A032741(k).

G.f.: exp(Sum_{k>=1} sigma_1(k)*x^(2*k)/(k*(1 - x^k))), where sigma_1(k) = sum of divisors of k (A000203).

A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 9, 5, 17, 17, 38, 33, 88, 75, 169, 181, 343, 353, 712, 728, 1348, 1518, 2591, 2898, 5025, 5615, 9259, 10866, 17160, 20111, 31775, 37264, 57130, 68782, 102663, 123698, 183793, 221708, 323077, 395325, 566079, 693248, 987086, 1210110, 1700074, 2100674, 2915549

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Convolution of A192065 and A255528.

Weigh transform of A001065.

N. J. A. Sloane, Transforms

Cf. A000203, A001065, A192065, A255528, A318784.

with(numtheory): a:=series(mul((1+x^k)^(sigma(k)-k),k=1..100),x=0,47): seq(coeff(a,x,n),n=0..46); # Paolo P. Lava, Apr 02 2019

nmax = 46; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 46; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[1, d] - 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 (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]

G.f.: Product_{k>=1} (1 + x^k)^A001065(k).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*(sigma_1(d) - d) ) * x^k/k).
a(n) ~ exp(-Pi^4 / (864*(Pi^2 - 6)*Zeta(3)) - Pi^2 * n^(1/3) / (12*(2*(Pi^2 - 6)*Zeta(3))^(1/3)) + 3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / 2^(5/3)) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (2^(17/24) * sqrt(3*Pi) * n^(2/3)). - Vaclav Kotesovec, Sep 11 2018

A321260 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(sigma_n(k)-k^n).

Original entry on oeis.org

1, 0, 1, 1, 18, 2, 861, 132, 106024, 40910, 72980055, 6838271, 228282942581, 27620223647, 2050169324675668, 352809815149813, 87174966874755673105, 6798293425492905407, 18318448554980083512011863, 1187839217207171380193247, 11258918803635775614062752424535

Offset: 0

Ilya Gutkovskiy, Nov 01 2018

nonn

Cf. A283333, A318783, A318784, A319647, A321258, A321261.

Table[SeriesCoefficient[Product[1/(1 - x^k)^(DivisorSigma[n, k] - k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^(2 k)/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

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

A326830 Expansion of Product_{i>=2, j>=2} 1 / (1 - x^(i*j))^j.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 5, 0, 9, 3, 17, 0, 46, 6, 68, 23, 153, 27, 297, 67, 534, 188, 978, 276, 1932, 620, 3250, 1313, 6033, 2246, 10854, 4361, 18776, 8639, 32831, 14835, 58230, 27635, 98052, 50980, 169522, 88243, 289720, 157179, 486232, 280206, 818006, 478014

Offset: 0

Ilya Gutkovskiy, Oct 20 2019

nonn

Euler transform of A048050.

Convolution of A326830 and A002865 is A318784. - Vaclav Kotesovec, Oct 26 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000203, A001001, A001157, A048050, A061256, A318784, A326831.

with(numtheory):
b:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 20 2019

nmax = 47; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

G.f.: Product_{k>=1} 1 / (1 - x^k)^A048050(k).
G.f.: exp(Sum_{k>=1} (A001001(k) - A000203(k) - A001157(k) + 1) * x^k / k).
a(n) ~ exp(3^(2/3) * ((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/2 - Pi^2 * (3/((Pi^2 - 6)*Zeta(3)))^(1/3) * n^(1/3)/4 - Pi^4 / (32*(Pi^2 - 6)*Zeta(3)) - 1/8) * A^(3/2)* (2*Pi)^(1/24) / (3^(1/8) * ((Pi^2 - 6)*Zeta(3))^(3/8) * n^(1/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 26 2019

A321262 Expansion of 1/(1 - Sum_{k>=1} kx^(2k)/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 1, 4, 3, 14, 12, 43, 50, 140, 177, 474, 643, 1560, 2325, 5246, 8194, 17763, 28838, 60190, 101063, 204935, 352227, 700037, 1224816, 2394971, 4250616, 8209174, 14724570, 28175997, 50949079, 96797183, 176131780, 332804667, 608449008, 1144920041, 2100793404

Offset: 0

Ilya Gutkovskiy, Nov 01 2018

nonn

Invert transform of A001065.

N. J. A. Sloane, Transforms

Cf. A001065, A180305, A318784, A319107, A320651, A321263.

a:=series(1/(1-add(k*x^(2*k)/(1-x^k),k=1..100)),x=0,38): seq(coeff(a,x,n),n=0..37); # Paolo P. Lava, Apr 02 2019

nmax = 37; CoefficientList[Series[1/(1 - Sum[k x^(2 k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[1/(1 - Sum[(k - EulerPhi[k]) x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(DivisorSigma[1, k] - k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 37}]

G.f.: 1/(1 - Sum_{k>=1} (sigma(k) - k)*x^k).
G.f.: 1/(1 - Sum_{k>=1} (k - phi(k))*x^k/(1 - x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A001065(k)*a(n-k).

cp's OEIS Frontend

A318783 Expansion of Product_{k>=1} 1/(1 - x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

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A319107 Expansion of Product_{k>=1} (1 + x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

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A321260 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(sigma_n(k)-k^n).

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A326830 Expansion of Product_{i>=2, j>=2} 1 / (1 - x^(i*j))^j.

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A321262 Expansion of 1/(1 - Sum_{k>=1} k*x^(2*k)/(1 - x^k)).

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A321262 Expansion of 1/(1 - Sum_{k>=1} kx^(2k)/(1 - x^k)).