A348082 -id:A348082 - 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)

A348081 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k^2 x).

Original entry on oeis.org

1, 5, 627, 251498, 209609235, 298201326150, 646748606934510, 1986821811445598260, 8209989926930833199235, 43919039258570117113742270, 295300365118450495520630242042, 2437724587984574697761809904387340, 24239364659088896670563082403144467630

Offset: 0

Seiichi Manyama, Sep 27 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..180

Cf. A036969, A234324, A269945, A298851, A348082.

Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)

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

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 78.52705817932973261726305432915417900827309581709564977985583533852704254... = (2+r)^6 / (r^2*(4+r)^2), where r = 0.329482909104375658581668801636329590897344... is the root of the equation 4+r = r*exp(6/(2+r)) and c = (2+r)/(Pi^(3/2)*sqrt(32 - 4*r*(4+r))) = 0.0815842039686253664272939415761688591712635596695065951780203519... - Vaclav Kotesovec, Oct 16 2021, updated May 17 2025
From Seiichi Manyama, May 13 2025: (Start)
a(n) = A036969(3*n,2*n) = A269945(3*n,2*n).
a(n) = (1/(4*n)!) * Sum_{k=0..4*n} (-1)^k * (2*n-k)^(6*n) * binomial(4*n,k).
a(n) = Sum_{k=0..2*n} (-2*n)^k * binomial(6*n,k) * Stirling2(6*n-k,4*n).
a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(2*n+k,2*n) * Stirling2(4*n-k,2*n). (End)

A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2k-1)^2 x).

Original entry on oeis.org

1, 1, 91, 24970, 14057043, 13444400190, 19558289594910, 40250341173506100, 111335096965772406915, 398473840263173643939190, 1791905773077609090895008106, 9890754761467721759394797416396, 65747198205879568307026776928408110

Offset: 0

Seiichi Manyama, Sep 28 2021

nonn

Cf. A001818, A298851, A348082, A348087, A381512.

Table[SeriesCoefficient[Product[1/(1 - (2*k-1)^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)

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

From Vaclav Kotesovec, Oct 16 2021, updated May 16 2025: (Start)
a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 52.447924272991536496097233490380538810534457762204101802471270109895148... and c = 0.028365099209561232079163758339093959048662789595134609351298413762...
In closed form, a(n) ~ 2^(2*n) * exp(2*n) * r^(n*r + 1/2) * (1+r)^(4*n) * n^(2*n - 1/2) / (sqrt(Pi*(1 - r*(2+r))) * (2+r)^((2+r)*n - 1/2)), where r = 0.044382033760833484984013906344747760869028157215190550759633... is the root of the equation exp(4/(1+r)) = (1 + 2/r). (End)
a(n) = A381512(n,2*n-1) = (1/(2^(2*n-2)*(2*n-1)!)) * Sum_{k=0..n-1} (-1)^k * (2*n-1-2*k)^(4*n-1) * binomial(2*n-1,k) for n > 0. - Seiichi Manyama, May 16 2025

cp's OEIS Frontend

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

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A348081 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k^2 x).

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A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2k-1)^2 x).

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cp's OEIS Frontend

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

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Author

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PARI

PARI

Formula

A348081 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - k^2 * x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2*k-1)^2 * x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

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

A348081 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k^2 x).

A348088 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2k-1)^2 x).