A032308 -id:A032308 - cp's OEIS frontend

A349924 Dirichlet g.f.: Product_{k>=2} (1 + 3 * k^(-s)).

1, 3, 3, 3, 3, 12, 3, 12, 3, 12, 3, 21, 3, 12, 12, 12, 3, 21, 3, 21, 12, 12, 3, 57, 3, 12, 12, 21, 3, 57, 3, 21, 12, 12, 12, 57, 3, 12, 12, 57, 3, 57, 3, 21, 21, 12, 3, 93, 3, 21, 12, 21, 3, 57, 12, 57, 12, 12, 3, 129, 3, 12, 21, 48, 12, 57, 3, 21, 12, 57

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

nonn

Cf. A032308, A045778, A339335, A347149, A349923.

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.

A370738 a(n) = 8^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/4).

Original entry on oeis.org

1, 6, -6, 1428, -13146, 280788, -3785820, 93142824, -1851272826, 37533646212, -765409050420, 16617464296728, -357906128318628, 7730398360992840, -168750405673899000, 3719099270015849040, -82288133754592611642, 1828585054153956768612, -40828782977534929747524, 915461326204911371035320

Offset: 0

Vaclav Kotesovec, Feb 28 2024

sign

Cf. A032308 (m=1), A370711 (m=2), A370712 (m=3), A370739 (m=5).

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

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

a(n) ~ (-1)^(n+1) * QPochhammer(-1/3)^(1/4) * 24^n / (4 * Gamma(3/4) * n^(5/4)).

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...

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)x^k) (1 + 3^(k-1)*x^k).

Original entry on oeis.org

1, 10, 39, 390, 1521, 7830, 49518, 207360, 951102, 4264650, 22185657, 89579520, 401428224, 1676401110, 7172977275, 31972081050, 130330236546, 537393139200, 2213787635712, 8988968449530, 36073295687070, 150459195064320, 590262148332288, 2362876271009370, 9314694641056095

Offset: 0

Vaclav Kotesovec, Mar 02 2024

nonn

In general, if d >= 1 and g.f. = Product_{k>=1} (1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k), then a(n) ~ d^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(d)^2))) * (Pi^2/3 + log(d)^2)^(1/4) / (2^(5/4) * sqrt(Pi) * (d+1) * n^(3/4)).

Cf. A032308, A344062.

Cf. A022567 (d=1), A370761 (d=2).

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

a(n) ~ 3^(n + 1/2) * exp(sqrt(2*n*(Pi^2/3 + log(3)^2))) * (Pi^2/3 + log(3)^2)^(1/4) / (2^(13/4) * sqrt(Pi) * n^(3/4)).

cp's OEIS Frontend

A349924 Dirichlet g.f.: Product_{k>=2} (1 + 3 * k^(-s)).

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Author

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370738 a(n) = 8^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)x^k) (1 + 3^(k-1)*x^k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A349924 Dirichlet g.f.: Product_{k>=2} (1 + 3 * k^(-s)).

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370738 a(n) = 8^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)*x^k) * (1 + 3^(k-1)*x^k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

A370792 Expansion of Product_{k>=1} (1 + 3^(k+1)x^k) (1 + 3^(k-1)*x^k).