A261567 -id:A261567 - cp's OEIS frontend

A261582 Expansion of Product_{k>=1} 1/(1 + 3*x^k).

1, -3, 6, -21, 69, -201, 591, -1785, 5406, -16194, 48426, -145380, 436641, -1309611, 3927399, -11783280, 35354139, -106059387, 318165729, -954506190, 2863556475, -8590643832, 25771817454, -77315531169, 231946940175, -695840583126, 2087520715788, -6262562872614

Offset: 0

Vaclav Kotesovec, Aug 25 2015

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Seiichi Manyama, Table of n, a(n) for n = 0..2094

Cf. A032308, A242587, A246935, A261565, A261567.

nmax = 40; CoefficientList[Series[Product[1/(1 + 3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*3^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 + 4/QPochhammer[-3, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) ~ c * (-3)^n, where c = Product_{j>=1} 1/(1-1/(-3)^j) = 1/QPochhammer[-1/3,-1/3] = 0.8212554466473167689981660621182786378...

G.f.: Sum_{i>=0} (-3)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

A261566 Expansion of Product_{k>=1} (1/(1 + 2*x^k))^k.

Original entry on oeis.org

1, -2, 0, -6, 16, -18, 48, -94, 208, -426, 752, -1646, 3360, -6578, 13056, -26358, 53456, -105890, 211392, -424366, 850544, -1699290, 3393136, -6795646, 13601184, -27188130, 54358000, -108752870, 217552976, -435033618, 869999584, -1740145118, 3480497584

Offset: 0

Vaclav Kotesovec, Aug 24 2015

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

Cf. A071109, A255528, A261567.

nmax = 40; CoefficientList[Series[Product[(1/(1 + 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*2^k/k*x^k/(1-x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1 - 1/(-2)^j)^(j+1) = 0.81033497534928929188778847125052151513524786804782471307090750707405...

A266857 Expansion of Product_{k>=1} (1 + 3*x^k)^k.

Original entry on oeis.org

1, 3, 6, 27, 48, 132, 324, 651, 1491, 3078, 6447, 12795, 25839, 50088, 96099, 184491, 343920, 640545, 1173609, 2138403, 3850584, 6882354, 12186336, 21423660, 37421757, 64816608, 111637392, 190976859, 324868530, 549265290, 923904711, 1545406077, 2572326510

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

In general, for m > 0, if g.f. = Product_{k>=1} (1 + m*x^k)^k then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (m+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(m) + log(m)^3 - 6*polylog(3, -1/m).

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

Cf. A026007, A261562, A261565, A261567.

nmax=50; CoefficientList[Series[Product[(1+3*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (2^(2/3) * 3^(2/3) * sqrt(Pi) * n^(2/3)), where c = Pi^2*log(3) + log(3)^3 - 6*polylog(3, -1/3) = 14.092743327504459346835224018840792668682349056875722467... .

A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

Original entry on oeis.org

1, -1, 0, -18, 208, -2400, 36504, -663754, 13808320, -324176418, 8487126400, -245122390601, 7741417124880, -265402847130421, 9816338228638872, -389618889514254225, 16518399076342421248, -745025763154442071130, 35619835529954597786208, -1799459812004380374518790, 95780758238408017088795600

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

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Cf. A255528, A261566, A261567, A266971, A292134, A297326, A298985, A298986, A298987.

Table[SeriesCoefficient[Product[1/(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) ~ (-1)^n * n^n * (1 - 2/n + 6/n^2 - 14/n^3 + 33/n^4 - 70/n^5 + 149/n^6 - 298/n^7 + 591/n^8 - 1132/n^9 + 2139/n^10 + ...), for coefficients, see A005380. - Vaclav Kotesovec, Aug 21 2018

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A261582 Expansion of Product_{k>=1} 1/(1 + 3*x^k).

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A261566 Expansion of Product_{k>=1} (1/(1 + 2*x^k))^k.

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A266857 Expansion of Product_{k>=1} (1 + 3*x^k)^k.

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A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

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