A300579 -id:A300579 - cp's OEIS frontend

A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

1, -1, -2, -7, -11, -43, -65, -259, -146, -1798, 826, -8116, 17593, -35089, 301903, -308464, 3582403, 157367, 28816009, 9388694, 329375419, -61352008, 2991009094, 509592773, 23675224255, 1207374806, 229200996508, -129896994130, 2090952547882, -816324790165, 14079091274800

Offset: 0

Ilya Gutkovskiy, Jun 08 2022

sign

Cf. A032308, A242587, A261582, A292128, A300579, A344062, A352402, A352786.

nmax = 30; CoefficientList[Series[Product[1/(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^k Length[IntegerPartitions[n, {k}]] 3^(n - k), {k, 0, n}], {n, 0, 30}]

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

A370338 Expansion of Product_{n>=1} (1 - 3^(n-1)x^n) (1 + 3^(n-1)*x^n)^2.

Original entry on oeis.org

1, 1, 2, 11, 24, 114, 297, 1224, 3240, 13230, 37017, 138510, 407754, 1469664, 4413366, 15717969, 47239200, 163408266, 511758000, 1719152586, 5348422224, 18083342907, 56672868240, 187301066040, 594207370746, 1947548449296, 6185182455792, 20263641256656, 64084643627283

Offset: 0

Paul D. Hanna, Feb 26 2024

nonn

Compare to Product_{n>=1} (1 - 3^n*x^n) * (1 + 3^n*x^n)^2 = Sum_{n>=0} 3^(n*(n+1)/2) * x^(n*(n+1)/2).

In general, for d > 1, if g.f. = Product_{k>=1} (1 - d^(k-1)*x^k) * (1 + d^(k-1)*x^k)^2, then a(n) ~ c^(1/4) * d^(n + 3/2) * exp(2*sqrt(c*n)) / (2 * sqrt((d-1)*Pi) * (d+1) * n^(3/4)), where c = -2*polylog(2, -1/d) - polylog(2, 1/d). - Vaclav Kotesovec, Feb 26 2024

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 24*x^4 + 114*x^5 + 297*x^6 + 1224*x^7 + 3240*x^8 + 13230*x^9 + 37017*x^10 + 138510*x^11 + 407754*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 3*x^2)*(1 + 3*x^2)^2 * (1 - 9*x^3)*(1 + 9*x^3)^2 * (1 - 27*x^4)*(1 + 27*x^4)^2 * ... * (1 - 3^(n-1)*x^n)*(1 + 3^(n-1)*x^n)^2 * ...
Compare A(x) to the series that results from a similar infinite product:
(1 - 3*x)*(1 + 3*x)^2 * (1 - 9*x^2)*(1 + 9*x^2)^2 * (1 - 27*x^3)*(1 + 27*x^3)^2 * (1 - 81*x^4)*(1 + 81*x^4)^2 * ... = 1 + 3*x + 27*x^3 + 729*x^6 + 59049*x^10 + 14348907*x^15 + 10460353203*x^21 + 22876792454961*x^28 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1035

Cf. A300579, A344062, A370337.

{a(n) = polcoeff( prod(k=1,n, (1 - 3^(k-1)*x^k) * (1 + 3^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,30, print1(a(n),", "))

a(n) ~ c^(1/4) * 3^(n + 3/2) * exp(2*sqrt(c*n)) / (2^(7/2) * sqrt(Pi) * n^(3/4)), where c = -2*polylog(2,-1/3) - polylog(2,1/3) = 0.2518530229985534570173197... - Vaclav Kotesovec, Feb 26 2024

A370711 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/2).

Original entry on oeis.org

1, 6, 6, 348, -570, 12084, -31332, 780792, -6111930, 65506884, -599418444, 6707736456, -69508986852, 738378468744, -7878832564872, 85524000547056, -929068361832378, 10158667075255524, -111690827626777788, 1234592278534799592, -13700571880245603276, 152613494540593338264

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032308, A370714.

Cf. A300579, A344062.

nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 3*(4*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-3, x]/4], {x, 0, nmax}], x] * 4^Range[0, nmax]

G.f.: Product_{k>=1} (1 + 3*(4*x)^k)^(1/2).

a(n) ~ (-1)^(n+1) * c * 12^n / n^(3/2), where c = QPochhammer(-1/3)^(1/2) / (2*sqrt(Pi)) = 0.311283382185276347775502154581850436407169685238...

A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

Original entry on oeis.org

1, 3, 0, 99, -270, 2430, -10287, 105462, -750141, 5702481, -42623901, 347424633, -2779077762, 22353287634, -181730796723, 1493711042589, -12321529794261, 102125312638713, -850797139405887, 7120067746384863, -59800770201017934, 503922807927384129, -4259721779079782751

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032308, A370710.

Cf. A300579, A344062.

nmax = 30; CoefficientList[Series[Product[(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), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(QPochhammer[-3, x]/4)^(1/3), {x, 0, nmax}], x] * 3^Range[0, nmax]

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

a(n) ~ (-1)^(n+1) * c * 9^n / n^(4/3), where c = QPochhammer(-1/3)^(1/3) / (3*Gamma(2/3)) = 0.26286302373105271371291957730496322329245126572...

A300581 Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).

Original entry on oeis.org

1, 4, 24, 112, 544, 2368, 10624, 44800, 190976, 791552, 3282944, 13414400, 54829056, 222117888, 899383296, 3625123840, 14601027584, 58659700736, 235555782656, 944552017920, 3786334535680, 15166305468416, 60736264994816, 243129089261568, 973133053952000

Offset: 0

Vaclav Kotesovec, Mar 09 2018

nonn

Cf. A075900, A300579.

Cf. A023882, A077365, A179381, A300520.

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

a(n) ~ c * 4^n, where c = A065446 = 1/QPochhammer(1/2) = 3.46274661945506361...

A323634 Expansion of Product_{k>=1} 1/(1 - k^(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 12, 80, 723, 8716, 128227, 2251086, 45647542, 1051845574, 27107414480, 772785074811, 24136982014698, 819697939365724, 30068912837398063, 1184872370227462528, 49914074776385885492, 2238476211786621770206, 106476394492364281869654, 5354276181476337307494676

Offset: 0

Ilya Gutkovskiy, Jan 21 2019

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = n^(n-1). - Seiichi Manyama, Aug 22 2020

Seiichi Manyama, Table of n, a(n) for n = 0..387

Cf. A023879, A023882, A266964, A299786, A300579.

a:=series(mul(1/(1-k^(k-1)*x^k),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019

nmax = 20; CoefficientList[Series[Product[1/(1 - k^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^(k - k/d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]

N=40; x='x+O('x^N); Vec(1/prod(k=1, N, 1-k^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020

a(n) ~ n^(n-1) * (1 + exp(-1)/n + (3*exp(-2) + 3*exp(-1)/2)/n^2). - Vaclav Kotesovec, Jan 22 2019

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.

cp's OEIS Frontend

A352762 Expansion of Product_{k>=1} 1 / (1 + 3^(k-1)*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370338 Expansion of Product_{n>=1} (1 - 3^(n-1)*x^n) * (1 + 3^(n-1)*x^n)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A370711 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370712 a(n) = 3^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A300581 Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A323634 Expansion of Product_{k>=1} 1/(1 - k^(k-1)*x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370338 Expansion of Product_{n>=1} (1 - 3^(n-1)x^n) (1 + 3^(n-1)*x^n)^2.