A348084 -id:A348084 - cp's OEIS frontend

A348085 a(n) = [x^n] Product_{k=1..2n} 1/(1 - (2k-1) * x).

1, 4, 170, 13776, 1652442, 262842580, 52116296024, 12380577235040, 3427841258566890, 1083931844930932140, 385417972804020879450, 152219732613102667656000, 66113646914860527721527960, 31319437721634527178263452656

Offset: 0

Seiichi Manyama, Sep 28 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..277

Cf. A039755, A293318, A348082, A348084, A348087.

a(n) = polcoef(1/prod(k=1, 2*n, 1-(2*k-1)*x+x*O(x^n)), n);

a(n) = if(n==0, 1, -sum(k=0, 2*n-1, (-1)^k*(2*k+1)^(3*n-1)*binomial(2*n-1, k))/(2^(2*n-1)*(2*n-1)!));

a(n) = A039755(3*n-1,2*n-1) for n > 0.
a(n) = (-1/(2^(2*n-1) * (2*n-1)!)) * Sum_{k=0..2*n-1} (-1)^k * (2*k+1)^(3*n-1) * binomial(2*n-1,k) for n > 0.
a(n) ~ 3^(3*n - 1/2) * n^(n - 1/2) / (sqrt(2*Pi*(1-c)) * (3 - 2*c)^n * c^(2*n - 1/2) * exp(n)), where c = -LambertW(-3*exp(-3/2)/2) = 0.62578253420128292... - Vaclav Kotesovec, Oct 02 2021
From Seiichi Manyama, May 16 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-1,k+2*n-1) * Stirling2(k+2*n-1,2*n-1) for n > 0.
a(n) = Sum_{k=0..n} (-2)^k * (4*n-1)^(n-k) * binomial(3*n-1,k+2*n-1) * Stirling2(k+2*n-1,2*n-1) for n > 0. (End)

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

Original entry on oeis.org

1, 6, 266, 22275, 2757118, 452329200, 92484925445, 22653141490980, 6466506598695390, 2108114165258886708, 772778072287000494520, 314641228029527540596455, 140880584836935832288402135, 68799366730032076856334789900, 36392216443342587869022660451080, 20728132932716479897744043460870000

Offset: 0

Vaclav Kotesovec, May 13 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A007820, A348084, A383882.

Cf. A217913.

[&+[Abs(StirlingSecond(4*n, 3*n))]: n in [0..15]]; // Vincenzo Librandi, May 21 2025

Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 3*n}], {x, 0, n}], {n, 0, 15}]
Table[StirlingS2[4*n, 3*n], {n, 0, 15}]
Table[SeriesCoefficient[(-1)^n/(Pochhammer[1 - 1/x, 3*n]*x^(3*n)), {x, 0, n}], {n, 0, 15}]

a(n) = Stirling2(4*n,3*n).

a(n) ~ (-1)^(3*n) * 4^(4*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 3^(3*n + 1/2) * w^(3*n) * (4/3 + w)^n), where w = LambertW(-4/(3*exp(4/3))).

A383882 a(n) = [x^n] Product_{k=1..4n} 1/(1 - kx).

Original entry on oeis.org

1, 10, 750, 106470, 22350954, 6220194750, 2157580085700, 896587036640680, 434225240080346858, 240175986308550372366, 149377949042637543000150, 103192471874508023383125750, 78394850841083734162487127720, 64957213308036504429927388238088, 58298851680969051596827194829579744

Offset: 0

Vaclav Kotesovec, May 13 2025

nonn

In general, for m>=1, Stirling2((m+1)*n, m*n) ~ (-1)^(m*n) * (m+1)^((m+1)*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w(m))) * exp(n) * m^(m*n + 1/2) * w(m)^(m*n) * (1 + 1/m + w(m))^n), where w(m) = LambertW(-(1 + 1/m)/exp(1 + 1/m)).

Cf. A007820 (m=1), A348084 (m=2), A383881 (m=3).
Cf. A217913.
Cf. A187646, A384129, A384130.

Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 4*n}], {x, 0, n}], {n, 0, 15}]
Table[StirlingS2[5*n, 4*n], {n, 0, 15}]
Table[SeriesCoefficient[1/(Pochhammer[1 - 1/x, 4*n]*x^(4*n)), {x, 0, n}], {n, 0, 15}]

a(n) = Stirling2(5*n,4*n).

a(n) ~ 5^(5*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 4^(4*n + 1/2) * w^(4*n) * (5/4 + w)^n), where w = LambertW(-5/(4*exp(5/4))).

A384129 Number of permutations of 3n objects with exactly 2n cycles.

Original entry on oeis.org

1, 3, 85, 4536, 357423, 37312275, 4853222764, 756111184500, 137272511800831, 28460103232088385, 6634460278534540725, 1717750737160208150400, 489078062391738506912340, 151874660255802127280374140, 51082995429153110239690350120, 18500755859447038660174079965500

Offset: 0

Seiichi Manyama, May 20 2025

Cf. A132393, A187646, A348084.

PARI
```
a(n) = abs(stirling(3*n, 2*n, 1));
```

a(n) = A132393(3*n,2*n) = |Stirling1(3*n,2*n)|.
a(n) = (3*n)! * [x^(3*n)] log(1 - x)^(2*n) / (2*n)!.
a(n) ~ 3^(4*n - 1/2) * w^(3*n) * n^(n - 1/2) / (sqrt(Pi*(w-1)) * 2^(2*n + 1/2) * exp(n) * (3*w-2)^n), where w = -LambertW(-1, -2*exp(-2/3)/3) = 1.4293552275170056487... - Vaclav Kotesovec, May 23 2025

cp's OEIS Frontend

A348085 a(n) = [x^n] Product_{k=1..2n} 1/(1 - (2k-1) * x).

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A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

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A383882 a(n) = [x^n] Product_{k=1..4n} 1/(1 - kx).

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A384129 Number of permutations of 3n objects with exactly 2n cycles.

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Formula

cp's OEIS Frontend

A348085 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - (2*k-1) * x).

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PARI

PARI

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A383881 a(n) = [x^n] Product_{k=1..3*n} 1/(1 - k*x).

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Magma

Mathematica

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A383882 a(n) = [x^n] Product_{k=1..4*n} 1/(1 - k*x).

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Mathematica

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A384129 Number of permutations of 3*n objects with exactly 2*n cycles.

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PARI

Formula

A348085 a(n) = [x^n] Product_{k=1..2n} 1/(1 - (2k-1) * x).

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

A383882 a(n) = [x^n] Product_{k=1..4n} 1/(1 - kx).

A384129 Number of permutations of 3n objects with exactly 2n cycles.