A318783 -id:A318783

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A318784

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

+10
5

1, 0, 1, 1, 4, 2, 11, 6, 25, 20, 56, 44, 139, 107, 283, 266, 619, 567, 1317, 1242, 2680, 2705, 5403, 5539, 10947, 11339, 21291, 23013, 41494, 45213, 79991, 88312, 151546, 170908, 284901, 324421, 532505, 611227, 981002, 1142000, 1797451, 2105773, 3268765, 3855050, 5889704, 7004451

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Convolution of A061256 and A073592.

Euler transform of A001065.

LINKS

Table of n, a(n) for n=0..45.

N. J. A. Sloane, Transforms

FORMULA

G.f.: Product_{k>=1} 1/(1 - x^k)^A001065(k).

G.f.: exp(Sum_{k>=1} sigma_2(k)*x^(2*k)/(k*(1 - x^k))), where sigma_2(k) = sum of squares of divisors of k (A001157).

a(n) ~ exp(3^(2/3) * c^(1/3) * n^(2/3)/2 - Pi^2 * n^(1/3) / (4 * 3^(2/3) * c^(1/3)) - Pi^4/(288*c) - 1/8) * A^(3/2) * c^(1/8) / (3^(5/8) * (2*Pi)^(11/24) * n^(5/8)), where c = (Pi^2 - 6)*Zeta(3) and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Sep 03 2018

MAPLE

with(numtheory):

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

(sigma(d)-d), d=divisors(j))*a(n-j), j=1..n)/n)

end:

seq(a(n), n=0..50); # Alois P. Heinz, Sep 03 2018

MATHEMATICA

nmax = 45; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 45; CoefficientList[Series[Exp[Sum[DivisorSigma[2, 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[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

CROSSREFS

Cf. A000203, A001065, A001157, A061256, A073592, A318783.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Sep 03 2018

STATUS

approved

A321260

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..20.

FORMULA

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

MATHEMATICA

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Nov 01 2018

STATUS

approved

A318844

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

+10
1

1, 0, 1, 1, 2, 2, 5, 4, 8, 10, 15, 17, 29, 31, 48, 60, 81, 99, 143, 167, 231, 287, 374, 460, 615, 740, 964, 1194, 1512, 1856, 2379, 2877, 3635, 4460, 5540, 6759, 8433, 10192, 12608, 15335, 18774, 22726, 27868, 33525, 40863, 49292, 59652, 71694, 86780, 103818, 125118, 149778, 179608

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Convolution of A081362 and A107742.

Weigh transform of A032741.

LINKS

Table of n, a(n) for n=0..52.

N. J. A. Sloane, Transforms

FORMULA

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

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

MAPLE

with(numtheory): a:=series(mul((1+x^k)^(tau(k)-1), k=1..100), x=0, 53): seq(coeff(a, x, n), n=0..52); # Paolo P. Lava, Apr 02 2019

MATHEMATICA

nmax = 52; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 52; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 1) x^k/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[0, d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

CROSSREFS

Cf. A000005, A000203, A032741, A081362, A107742, A318783.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Sep 04 2018

STATUS

approved

A327798

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

+10
1

1, 0, 1, 1, 3, 3, 9, 10, 25, 34, 72, 106, 215, 330, 635, 1025, 1899, 3141, 5713, 9602, 17213, 29292, 51982, 89149, 157249, 271027, 476037, 823386, 1442063, 2500015, 4370386, 7588146, 13248591, 23026728, 40169991, 69865026, 121811765, 211954826, 369412910

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Invert transform of A032741.

LINKS

Robert Israel, Table of n, a(n) for n = 0..3000

FORMULA

G.f.: 1 / (1 - Sum_{k>=1} x^(2*k) / (1 - x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A032741(k) * a(n-k).

MAPLE

N:= 100: # for a(0)..a(N)

G:= 1/(1-add(x^(2*k)/(1-x^k), k=1..(N+1)/2)):

S:= series(G, x, N+1):

seq(coeff(S, x, i), i=0..N); # Robert Israel, Jan 10 2023

MATHEMATICA

nmax = 38; CoefficientList[Series[1/(1 - Sum[x^(2 k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

a[0] = 1; a[n_] := a[n] = Sum[(DivisorSigma[0, k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 38}]

CROSSREFS

Cf. A032741, A129921, A318783, A327739.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Sep 25 2019

STATUS

approved

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