A242587 -id:A242587 - cp's OEIS frontend

A266944 Expansion of Product_{k>=1} 1 / (1 - 3*x^k)^2.

1, 6, 33, 150, 636, 2508, 9501, 34674, 123369, 429396, 1469733, 4959600, 16545597, 54662046, 179124837, 582893052, 1885479918, 6067245570, 19435083054, 62006825166, 197128631562, 624716063502, 1974151076946, 6222482535642, 19567579430643, 61403207075448

Offset: 0

Vaclav Kotesovec, Jan 06 2016

nonn

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

Cf. A242587, A266943.

nmax = 40; CoefficientList[Series[Product[1/(1-3*x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * n * 3^n, where c = Product_{k>=1} 1/(1-1/3^k)^2 = 1/QPochhammer(1/3)^2 = 3.187340158492291107944103748176139... .

A349922 Dirichlet g.f.: Product_{k>=2} 1 / (1 - 3 * k^(-s)).

Original entry on oeis.org

1, 3, 3, 12, 3, 12, 3, 39, 12, 12, 3, 48, 3, 12, 12, 129, 3, 48, 3, 48, 12, 12, 3, 165, 12, 12, 39, 48, 3, 57, 3, 399, 12, 12, 12, 201, 3, 12, 12, 165, 3, 57, 3, 48, 48, 12, 3, 552, 12, 48, 12, 48, 3, 165, 12, 165, 12, 12, 3, 237, 3, 12, 48, 1245, 12, 57, 3, 48, 12, 57

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

nonn

Cf. A001055, A165824, A242587, A339318, A349921.

A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/3).

Original entry on oeis.org

1, 6, 36, 360, 1998, 18792, 121176, 1123632, 7537860, 72078174, 510702408, 4896308088, 35923749480, 345406994280, 2600934294816, 24985346997888, 191735328374478, 1838307293836560, 14317601666954364, 136953233511162840, 1079293961918593800, 10299943344889922832

Offset: 0

Vaclav Kotesovec, Feb 29 2024

nonn

In general, if d > 1, m >= 1 and g.f. = Product_{k>=1} ((1 + d*x^k)/(1 - d*x^k))^(1/m), then a(n) ~ QPochhammer(-1, 1/d)^(1/m) * d^n / (Gamma(1/m) * QPochhammer(1/d)^(1/m) * n^(1 - 1/m)).

Cf. A303390 (d=3,m=1), A370751 (d=3,m=2), A370752 (d=3,m=3).
Cf. A261584 (d=2,m=1), A303346 (d=2,m=2), A370750 (d=2,m=3), A370749 (d=2,m=4).
Cf. A015128 (d=1,m=1), A303307 (d=1,m=2), A303342 (d=1,m=3).
Cf. A032308, A242587.
Cf. A370712, A370710.

nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k)/(1 - 3*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(3*x)^k)/(1 - 3*(3*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + 3*(3*x)^k)/(1 - 3*(3*x)^k))^(1/3).

a(n) ~ QPochhammer(-1, 1/3)^(1/3) * 9^n / (Gamma(1/3) * QPochhammer(1/3)^(1/3) * n^(2/3)).

A300583 Expansion of Product_{k>=1} 1 / (1 - 23^kx^k).

Original entry on oeis.org

1, 6, 54, 378, 2754, 17982, 121014, 765450, 4894506, 30429918, 189311094, 1160312850, 7113869226, 43228473822, 262556300286, 1587419581410, 9590551158474, 57795130268694, 348125978482686, 2093918636332530, 12590534397102930, 75647788993941174

Offset: 0

Vaclav Kotesovec, Mar 09 2018

Cf. A065446, A242587, A300582.

nmax = 25; CoefficientList[Series[Product[1/(1-2*3^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 6^n, where c = A065446 = 1/QPochhammer(1/2) = 3.46274661945506361153...

A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

Original entry on oeis.org

1, -1, -3, -6, -18, -27, -108, -81, -486, 0, -1458, 8748, -6561, 118098, 118098, 1003833, 1417176, 11691702, 9565938, 105225318, 114791256, 746143164, 1076168025, 7231849128, 2324522934, 58113073350, 45328197213, 334731302496, 146444944842, 3263630199336, -3012581722464

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A003056, A032308, A242587, A261582, A292128, A300579, A337299, A344062, A352762.

nmax = 30; CoefficientList[Series[Product[(1 - 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 30}]

a(n) = Sum_{k=0..A003056(n)} (-1)^k * q(n,k) * 3^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

A370734 a(n) = 8^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/4).

Original entry on oeis.org

1, 6, 138, 2292, 47046, 852756, 18266628, 366635112, 7948637382, 170568754692, 3761729402412, 83136335360856, 1863229219846428, 41883396293989320, 948524060727094728, 21555960625992644304, 492036151405623971142, 11264431786398948383844, 258676355450246122857756

Offset: 0

Vaclav Kotesovec, Feb 28 2024

nonn

Cf. A242587 (m=1), A370714 (m=2), A370710 (m=3), A370735 (m=5).

nmax = 20; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 8^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1/(1-3*(8*x)^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - 3*(8*x)^k)^(1/4).

a(n) ~ 24^n / (Gamma(1/4) * QPochhammer(1/3)^(1/4) * n^(3/4)).

A370751 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/2).

Original entry on oeis.org

1, 6, 30, 204, 966, 5748, 29388, 169944, 886278, 5169732, 27794820, 162920616, 894445212, 5274022920, 29398573272, 174041671344, 980746798278, 5821525480164, 33071756442708, 196663513473672, 1124154722216244, 6693497121210648, 38448301937075112, 229149691659210192

Offset: 0

Vaclav Kotesovec, Feb 29 2024

nonn

Cf. A303390, A370752.
Cf. A032308, A242587.
Cf. A370711, A370714.

nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k)/(1 - 3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(2*x)^k)/(1 - 3*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + 3*(2*x)^k)/(1 - 3*(2*x)^k))^(1/2).

a(n) ~ c * 6^n / n^(1/2), where c = (QPochhammer(-1,1/3) / (Pi * QPochhammer(1/3)))^(1/2) = 1.333660169175690343841707335109800906849893636...

A329155 Expansion of Product_{k>=1} 1 / (1 - 2x^k - 3x^(2*k))^(1/2).

Original entry on oeis.org

1, 1, 4, 9, 27, 67, 193, 515, 1462, 4070, 11588, 32898, 94389, 271017, 782401, 2263002, 6565987, 19086043, 55597255, 162207806, 473992799, 1386875848, 4062919108, 11915397853, 34979609583, 102781548770, 302259362326, 889566748760, 2619915414564, 7721166976185

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

nonn

Cf. A002426, A067855, A081362, A242587.

nmax = 29; CoefficientList[Series[Product[1/(1 - 2 x^k - 3 x^(2 k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[Sum[(3^d + (-1)^d)/d, {d, Divisors[k]}] x^k/2, {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 - x^(2*k - 1)) / (1 - 3*x^k))^(1/2).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (3^d + (-1)^d) / d ) * x^k / 2).
G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A002426 (central trinomial coefficients).
a(n) ~ c * 3^(n + 1/2) / (2*sqrt(Pi*n)), where c = sqrt(Product_{k>=2} 1/((1 - 1/3^(k-1))*(1 + 1/3^k))) = sqrt(8 / (3 * QPochhammer[-1, 1/3] * QPochhammer[1/3])) = 1.23332761652608605487734981242239445... - Vaclav Kotesovec, Nov 07 2019

cp's OEIS Frontend

A266944 Expansion of Product_{k>=1} 1 / (1 - 3*x^k)^2.

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A349922 Dirichlet g.f.: Product_{k>=2} 1 / (1 - 3 * k^(-s)).

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A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3*x^k)/(1 - 3*x^k))^(1/3).

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A300583 Expansion of Product_{k>=1} 1 / (1 - 2*3^k*x^k).

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A352786 Expansion of Product_{k>=1} (1 - 3^(k-1)*x^k).

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A370734 a(n) = 8^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/4).

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A370751 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 3*x^k)/(1 - 3*x^k))^(1/2).

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A329155 Expansion of Product_{k>=1} 1 / (1 - 2*x^k - 3*x^(2*k))^(1/2).

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A370752 a(n) = 3^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/3).

A300583 Expansion of Product_{k>=1} 1 / (1 - 23^kx^k).

A370751 a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 3x^k)/(1 - 3x^k))^(1/2).

A329155 Expansion of Product_{k>=1} 1 / (1 - 2x^k - 3x^(2*k))^(1/2).