A309319 -id:A309319 - cp's OEIS frontend

A088994 Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.

1, 1, 0, 2, 8, 24, 144, 720, 8448, 64512, 576000, 5529600, 74972160, 887546880, 11285084160, 168318259200, 2843121254400, 44790578380800, 747955947110400, 13937735643955200, 287117441217331200, 5838778006909747200, 120976472421826560000, 2712639152754878054400

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 01 2003

nonn

a(n) is the number of n-permutations composed only of odd cycles of distinct length. - Geoffrey Critzer, Mar 08 2013

Also the number of permutations p of [n] with unique (functional) square root, i.e., there exists a unique permutation g such that g^2 = p. - Keith J. Bauer, Jan 08 2024

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000142, A007838, A007841, A087639, A088335, A294506, A305199, A309319.

b:= proc(n, i) option remember; `if`(((i+1)/2)^2n, 0, (i-1)!*
       b(n-i, i-2)*binomial(n, i))))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 01 2017

nn=20;Range[0,nn]!CoefficientList[Series[Product[1+x^(2i-1)/(2i-1),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Mar 08 2013 *)

{a(n)=n!*polcoeff( prod(k=1, n, 1+(k%2)*x^k/k, 1+x*O(x^n)), n)} /* Michael Somos, Sep 19 2006 */

E.g.f.: Product_{m >= 1} (1+x^(2*m-1)/(2*m-1)). - Vladeta Jovovic, Nov 05 2003
a(n) ~ exp(-gamma/2) * n! / sqrt(2*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019
a(n) = n! - A088335(n). - Alois P. Heinz, Jan 27 2020

More terms from Vladeta Jovovic, Nov 03 2003

a(0)=1 prepended by Seiichi Manyama, Nov 01 2017

A294506 E.g.f.: 1/Product_{k>0} (1-x^(2k-1)/(2k-1)).

Original entry on oeis.org

1, 1, 2, 8, 32, 184, 1184, 9008, 74752, 726528, 7583232, 87931392, 1092516864, 14863589376, 215094226944, 3358032635904, 55181218873344, 970561417248768, 17945595514847232, 351221170194874368, 7186120683011702784, 155103171658691641344

Offset: 0

Seiichi Manyama, Nov 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 5).

Cf. A007838, A007841, A087639, A088994, A309319.

nmax = 30; CoefficientList[Series[1/Product[(1-x^(2*k-1)/(2*k-1)), {k, 1, Floor[nmax/2] + 1}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 02 2017 *)

a(n) ~ 2*exp(-gamma/2) * sqrt(2*n) * n! / Pi, where gamma is the Euler-Mascheroni constant A001620 [Lehmer, 1972]. - Vaclav Kotesovec, Jul 23 2019

A326755 E.g.f.: Product_{k>=1} 1/(1 - x^(3k-1)/(3k-1)).

Original entry on oeis.org

1, 0, 1, 0, 6, 24, 90, 504, 7560, 18144, 485352, 4626720, 32033232, 516559680, 9142044912, 64700161344, 1804378343040, 29722011830784, 308081755013760, 8202581858225664, 184073277074529024, 2067986628774743040, 75069447974837132544, 1673053361596502645760

Offset: 0

Vaclav Kotesovec, Jul 23 2019

nonn

In the article by Lehmer, Theorem 7, p. 387, case b <> 0 and b <> 1, correct formula is W_n(S_a,b) ~ a^(-1/a) * exp(-gamma/a) * (Gamma((b-1)/a) / (Gamma(b/a) * Gamma(1/a))) * n^(1/a - 1).

Vaclav Kotesovec, Table of n, a(n) for n = 0..448
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).

Cf. A007841, A294506, A309319, A326756.

nmax = 25; CoefficientList[Series[1/Product[(1-x^(3*k-1)/(3*k-1)), {k, 1, Floor[nmax/3] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 3^(1/6) * exp(-gamma/3) * Gamma(1/3) * n! / (2*Pi*n^(2/3)).

a(n) ~ exp(-gamma/3) * n! / (3^(1/3) * Gamma(2/3) * n^(2/3)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

A326756 E.g.f.: Product_{k>=1} 1/(1 - x^(3k-2)/(3k-2)).

Original entry on oeis.org

1, 1, 2, 6, 30, 150, 900, 7020, 58680, 528120, 5644080, 63510480, 769610160, 10483933680, 150733677600, 2272680828000, 37752297264000, 653710445308800, 11839468023187200, 231623795388268800, 4723930089495302400, 99779582243860358400, 2249431677071465356800

Offset: 0

Vaclav Kotesovec, Jul 23 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..447
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7).

Cf. A007841, A294506, A309319, A326755.

nmax = 25; CoefficientList[Series[1/Product[(1-x^(3*k-2)/(3*k-2)), {k, 1, Floor[nmax/3] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 3^(5/3) * exp(-gamma/3) * n^(1/3) * n! / Gamma(1/3)^2, where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function [Lehmer, 1972]. - Vaclav Kotesovec, Jul 23 2019

A087639 E.g.f.: Product_{m >= 1} (1+x^(2m)/(2m)) (even powers only).

Original entry on oeis.org

1, 1, 6, 210, 8400, 740880, 88814880, 15217282080, 3319002086400, 992431440000000, 351841557779712000, 156995673442223616000, 82429416503416958976000, 52017974139195896832000000, 37547796668359538444083200000, 31987697744989345038846566400000

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 06 2003

nonn

Number of permutations of 2*n elements with distinct cycle lengths and without odd cycles. - Vladeta Jovovic, Aug 17 2004

Alois P. Heinz, Table of n, a(n) for n = 0..225

Cf. A007838, A007841, A088994, A294506, A305199, A309319.

b:= proc(n, i) option remember; `if`((i/2)*(i/2+1)n, 0, (i-1)!*
       b(n-i, i-2)*binomial(n, i))))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, Nov 01 2017

nmax = 20; Table[(CoefficientList[Series[Product[1 + x^(2*k)/(2*k), {k, 1, 2*nmax}], {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[2*n + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Jul 23 2019 *)

a(n) ~ 2*exp(-gamma/2) * (2*n)! / (Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019

More terms from Christian G. Bower, Jan 06 2006

A326779 E.g.f.: Product_{k>=1} 1/(1 - x^(4k-1)/(4k-1)).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 80, 720, 0, 13440, 172800, 3628800, 5913600, 98841600, 4420915200, 92559667200, 110702592000, 6012444672000, 234205087334400, 6616915329024000, 13708373852160000, 771938716483584000, 40374130262409216000, 1172555787961958400000

Offset: 0

Vaclav Kotesovec, Jul 24 2019

nonn

In the article by Lehmer, Theorem 7, p. 387, case b <> 0 and b <> 1, correct formula is W_n(S_a,b) ~ a^(-1/a) * exp(-gamma/a) * (Gamma((b-1)/a) / (Gamma(b/a) * Gamma(1/a))) * n^(1/a - 1), where gamma is the Euler-Mascheroni constant (A001620) and Gamma() is the Gamma function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..449
Vaclav Kotesovec, Graph - the asymptotic ratio (50000 terms)
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).

Cf. A007841, A294506, A309319, A326755, A326756, A326780.

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

a(n) ~ exp(-gamma/4) * n! / (2 * sqrt(Pi) * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.

A326780 E.g.f.: Product_{k>=1} 1/(1 - x^(4k-3)/(4k-3)).

Original entry on oeis.org

1, 1, 2, 6, 24, 144, 864, 6048, 48384, 475776, 4902912, 53932032, 647184384, 8892398592, 126430875648, 1906924529664, 30510792474624, 539606261956608, 9890452422918144, 188459240926150656, 3773077461736095744, 81667528704634650624, 1819516013302975561728

Offset: 0

Vaclav Kotesovec, Jul 24 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..447
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7).

Cf. A007841, A294506, A309319, A326755, A326756, A326779.

nmax = 25; CoefficientList[Series[1/Product[(1-x^(4*k-3)/(4*k-3)), {k, 1, Floor[nmax/4] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ 2^(7/2) * exp(-gamma/4) * n^(1/4) * n! / Gamma(1/4)^2, where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function [Lehmer, 1972].

cp's OEIS Frontend

A088994 Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.

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A294506 E.g.f.: 1/Product_{k>0} (1-x^(2*k-1)/(2*k-1)).

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A326755 E.g.f.: Product_{k>=1} 1/(1 - x^(3*k-1)/(3*k-1)).

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A326756 E.g.f.: Product_{k>=1} 1/(1 - x^(3*k-2)/(3*k-2)).

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A087639 E.g.f.: Product_{m >= 1} (1+x^(2*m)/(2*m)) (even powers only).

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Maple

Mathematica

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A326779 E.g.f.: Product_{k>=1} 1/(1 - x^(4*k-1)/(4*k-1)).

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Comments

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Crossrefs

Programs

Mathematica

Formula

A326780 E.g.f.: Product_{k>=1} 1/(1 - x^(4*k-3)/(4*k-3)).

Views

Author

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Links

Crossrefs

Programs

Mathematica

Formula

A294506 E.g.f.: 1/Product_{k>0} (1-x^(2k-1)/(2k-1)).

A326755 E.g.f.: Product_{k>=1} 1/(1 - x^(3k-1)/(3k-1)).

A326756 E.g.f.: Product_{k>=1} 1/(1 - x^(3k-2)/(3k-2)).

A087639 E.g.f.: Product_{m >= 1} (1+x^(2m)/(2m)) (even powers only).

A326779 E.g.f.: Product_{k>=1} 1/(1 - x^(4k-1)/(4k-1)).

A326780 E.g.f.: Product_{k>=1} 1/(1 - x^(4k-3)/(4k-3)).