A344064 -id:A344064 - cp's OEIS frontend

A344062 Expansion of Product_{k>=1} (1 + 3^(k-1)*x^k).

1, 1, 3, 12, 36, 135, 432, 1539, 4860, 17496, 55404, 192456, 623295, 2125764, 6849684, 23442453, 75110328, 252965916, 822670668, 2735858268, 8838926712, 29501352792, 95090206689, 314068876416, 1018141045092, 3342663979092, 10798571289897, 35481518064576

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

Cf. A003056, A008289, A032308, A300579, A304961, A340103, A344063, A344064, A344065, A344066, A344067, A344068.

nmax = 27; CoefficientList[Series[Product[(1 + 3^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 3^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 27}]

seq(n)={Vec(prod(k=1, n, 1 + 3^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

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

a(n) ~ (-polylog(2, -1/3))^(1/4) * 3^n * exp(2*sqrt(-polylog(2, -1/3)*n)) / (4*sqrt(Pi/3)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344063 Expansion of Product_{k>=1} (1 + 4^(k-1)*x^k).

Original entry on oeis.org

1, 1, 4, 20, 80, 384, 1600, 7424, 30720, 143360, 593920, 2703360, 11403264, 51118080, 214958080, 965738496, 4047503360, 17951621120, 76168560640, 334202142720, 1411970498560, 6211596451840, 26203595472896, 114246130073600, 484815908372480, 2101441598586880, 8896148580335616

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

Cf. A003056, A008289, A261568, A304961, A338673, A340103, A344062, A344064, A344065, A344066, A344067, A344068.

nmax = 26; CoefficientList[Series[Product[(1 + 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 4^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 26}]

seq(n)={Vec(prod(k=1, n, 1 + 4^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 4^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/4))^(1/4) * 4^n * exp(2*sqrt(-polylog(2, -1/4)*n)) / (2*sqrt(5*Pi/4)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344065 Expansion of Product_{k>=1} (1 + 6^(k-1)*x^k).

Original entry on oeis.org

1, 1, 6, 42, 252, 1728, 10584, 71280, 435456, 2939328, 17962560, 119532672, 739031040, 4867527168, 30051689472, 198147658752, 1221537687552, 7984437608448, 49643697954816, 321998350270464, 1997815999463424, 12977575759282176, 80455233450737664, 519208351807832064

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

Cf. A003056, A008289, A304961, A338675, A340103, A344062, A344063, A344064, A344066, A344067, A344068.

nmax = 23; CoefficientList[Series[Product[(1 + 6^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 6^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 23}]

seq(n)={Vec(prod(k=1, n, 1 + 6^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 6^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/6))^(1/4) * 6^n * exp(2*sqrt(-polylog(2, -1/6)*n)) / (2*sqrt(7*Pi/6)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344066 Expansion of Product_{k>=1} (1 + 7^(k-1)*x^k).

Original entry on oeis.org

1, 1, 7, 56, 392, 3087, 21952, 170471, 1210104, 9411920, 66824632, 513890832, 3683707839, 28086110472, 201122377288, 1534688027817, 10978118077136, 83158453503608, 599161640356888, 4508826988300152, 32435340235930576, 244366486039786096, 1756858874561956865, 13161303959340223232

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

Cf. A003056, A008289, A304961, A338676, A340103, A344062, A344063, A344064, A344065, A344067, A344068.

nmax = 23; CoefficientList[Series[Product[(1 + 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 7^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 23}]

seq(n)={Vec(prod(k=1, n, 1 + 7^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 7^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/7))^(1/4) * 7^n * exp(2*sqrt(-polylog(2, -1/7)*n)) / (4*sqrt(2*Pi/7)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344067 Expansion of Product_{k>=1} (1 + 8^(k-1)*x^k).

Original entry on oeis.org

1, 1, 8, 72, 576, 5120, 41472, 364544, 2949120, 25952256, 209977344, 1830813696, 14931722240, 129251672064, 1053340729344, 9123584278528, 74294344286208, 639503450505216, 5239722662166528, 44846880273727488, 367008185258606592, 3144110674230116352, 25718087147075928064

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

Cf. A003056, A008289, A304961, A338677, A340103, A344062, A344063, A344064, A344065, A344066, A344068.

nmax = 22; CoefficientList[Series[Product[(1 + 8^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 8^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 22}]

seq(n)={Vec(prod(k=1, n, 1 + 8^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 8^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/8))^(1/4) * 8^n * exp(2*sqrt(-polylog(2, -1/8)*n)) / (6*sqrt(Pi/8)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A344068 Expansion of Product_{k>=1} (1 + 9^(k-1)*x^k).

Original entry on oeis.org

1, 1, 9, 90, 810, 8019, 72900, 715149, 6495390, 63772920, 579270690, 5643903420, 51613018479, 499772430810, 4567687565310, 44250780833091, 404188047763920, 3894703308072990, 35764052204589030, 342923118899865390, 3146016498406236720, 30187757787717436380, 276843069234653897241

Offset: 0

Ilya Gutkovskiy, May 08 2021

nonn

In general, if g.f. = Product_{k>=1} (1 + d^(k-1)*x^k), where d > 1, then a(n) ~ (-polylog(2, -1/d))^(1/4) * d^n * exp(2*sqrt(-polylog(2, -1/d)*n)) / (2*sqrt((1 + 1/d)*Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Cf. A003056, A008289, A304961, A338678, A340103, A344062, A344063, A344064, A344065, A344066, A344067.

nmax = 22; CoefficientList[Series[Product[(1 + 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 9^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 22}]

seq(n)={Vec(prod(k=1, n, 1 + 9^(k-1)*x^k + O(x*x^n)))} \\ Andrew Howroyd, May 08 2021

a(n) = Sum_{k=0..A003056(n)} q(n,k) * 9^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ (-polylog(2, -1/9))^(1/4) * 9^n * exp(2*sqrt(-polylog(2, -1/9)*n)) / (2*sqrt(10*Pi/9)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A370435 Expansion of Product_{n>=1} (1 - 5^(n-1)x^n) (1 + 5^(n-1)*x^n)^2.

Original entry on oeis.org

1, 1, 4, 29, 120, 820, 3625, 23400, 105000, 669500, 3075625, 18837500, 89237500, 532500000, 2554062500, 15086640625, 72843750000, 421773437500, 2084812500000, 11834804687500, 58638281250000, 332210205078125, 1656773437500000, 9240966796875000, 46624682617187500, 257479980468750000

Offset: 0

Paul D. Hanna, Feb 26 2024

nonn

Compare to Product_{n>=1} (1 - 5^n*x^n) * (1 + 5^n*x^n)^2 = Sum_{n>=0} 5^(n*(n+1)/2) * x^(n*(n+1)/2).

			G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 120*x^4 + 820*x^5 + 3625*x^6 + 23400*x^7 + 105000*x^8 + 669500*x^9 + 3075625*x^10 + 18837500*x^11 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 5*x^2)*(1 + 5*x^2)^2 * (1 - 25*x^3)*(1 + 25*x^3)^2 * (1 - 125*x^4)*(1 + 125*x^4)^2 * ... * (1 - 5^(n-1)*x^n)*(1 + 5^(n-1)*x^n)^2 * ...

Paul D. Hanna, Table of n, a(n) for n = 0..630

Cf. A370337, A370338, A370434, A344064, A338674.

{a(n) = polcoeff( prod(k=1,n, (1 - 5^(k-1)*x^k) * (1 + 5^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,40, print1(a(n),", "))

a(n) ~ c^(1/4) * 5^(n + 3/2) * exp(2*sqrt(c*n)) / (24 * sqrt(Pi) * n^(3/4)), where c = -2*polylog(2, -1/5) - polylog(2, 1/5). - Vaclav Kotesovec, Feb 27 2024

cp's OEIS Frontend

A344062 Expansion of Product_{k>=1} (1 + 3^(k-1)*x^k).

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A344063 Expansion of Product_{k>=1} (1 + 4^(k-1)*x^k).

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A344065 Expansion of Product_{k>=1} (1 + 6^(k-1)*x^k).

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A344066 Expansion of Product_{k>=1} (1 + 7^(k-1)*x^k).

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A344067 Expansion of Product_{k>=1} (1 + 8^(k-1)*x^k).

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A344068 Expansion of Product_{k>=1} (1 + 9^(k-1)*x^k).

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A370435 Expansion of Product_{n>=1} (1 - 5^(n-1)*x^n) * (1 + 5^(n-1)*x^n)^2.

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A370435 Expansion of Product_{n>=1} (1 - 5^(n-1)x^n) (1 + 5^(n-1)*x^n)^2.