A303390 -id:A303390 - cp's OEIS frontend

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

1, 8, 40, 200, 872, 3720, 15400, 62920, 254440, 1024648, 4112680, 16483400, 66000360, 264150920, 1056903080, 4228272200, 16914393832, 67660396040, 270647139240, 1082600410440, 4330424811880, 17321748357640, 69287088965800, 277148557003720, 1108594618342760

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..1600

Cf. A261568, A246936, A067855, A303350.

Cf. A015128, A261584, A303390.

N:= 50: # for a(0)..a(N)
G:= mul((1+4*x^k)/(1-4*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 13 2019

nmax = 25; CoefficientList[Series[Product[(1+4*x^k)/(1-4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = QPochhammer[-1, 1/4] / QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

A303398 Expansion of Product_{k>=1} (1 - 3x^k)/(1 + 3x^k).

Original entry on oeis.org

1, -6, 12, -24, 102, -312, 840, -2544, 7788, -23406, 69816, -208968, 628536, -1886712, 5654784, -16961856, 50900934, -152709936, 458084244, -1374231912, 4122828408, -12368549040, 37105252680, -111315549552, 333947845416, -1001844169854, 3005528872008

Offset: 0

Seiichi Manyama, Apr 23 2018

sign

Robert Israel, Table of n, a(n) for n = 0..2094

Expansion of Product_{k>=1} (1 - b*x^k)/(1 + b*x^k): A002448 (b=1), A303397 (b=2), this sequence (b=3), A303402 (b=4).

Cf. A303390.

N:= 100: # for a(0)..a(N)
G:= mul((1-3*x^k)/(1+3*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Jul 31 2020

nmax = 30; CoefficientList[Series[Product[(1 - 3*x^k)/(1 + 3*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-3*x^k)/(1+3*x^k)))

a(n) ~ c * (-3)^n, where c = QPochhammer[-1, -1/3]/QPochhammer[-1/3] = 1.1824106844873309732830080836112464096086... - Vaclav Kotesovec, Apr 25 2018

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)).

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...

cp's OEIS Frontend

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

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A303398 Expansion of Product_{k>=1} (1 - 3x^k)/(1 + 3x^k).

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

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

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cp's OEIS Frontend

A303391 Expansion of Product_{k>=1} (1 + 4*x^k)/(1 - 4*x^k).

Views

Author

Keywords

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Programs

Maple

Mathematica

Formula

A303398 Expansion of Product_{k>=1} (1 - 3*x^k)/(1 + 3*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

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

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

A303398 Expansion of Product_{k>=1} (1 - 3x^k)/(1 + 3x^k).

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

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