A305199 -id:A305199 - 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

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

A326859 E.g.f.: Product_{k>=1} (1 + x^(2k-1)/(2k-1)) / (1 - x^(2k-1)/(2k-1)).

Original entry on oeis.org

1, 2, 4, 16, 80, 448, 3136, 24512, 218368, 2169600, 23752704, 282882048, 3661787136, 51076644864, 763127316480, 12184024793088, 206679198990336, 3710945850949632, 70379726872117248, 1405297582300200960, 29462283451103182848, 647386758393297371136

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Cf. A088994, A294506, A305199.

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

a(n) ~ exp(-gamma) * n * n!, where gamma is the Euler-Mascheroni constant A001620.

A326860 E.g.f.: Product_{k>=1} (1 + x^(3k-1)/(3k-1)) / (1 - x^(3k-1)/(3k-1)).

Original entry on oeis.org

1, 0, 2, 0, 12, 48, 180, 2016, 15120, 72576, 1424304, 11249280, 113164128, 2066238720, 22751977248, 303261573888, 6400216892160, 85934653249536, 1440131337066240, 34330891188013056, 549029461368181248, 11212163885207900160, 296439802585781976576

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

In general, if c > 0, mod(d,c) <> 0, mod(d,c) <> 1 and e.g.f. = Product_{k>=1} (1 + x^(c*k+d)/(c*k+d)) / (1 - x^(c*k+d)/(c*k+d)), then a(n) ~ Gamma(1 + (d-1)/c) * n^(2/c - 1) * n! / (c^(2/c) * exp(2*gamma/c) * Gamma(2/c) * Gamma(1 + (d+1)/c)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

Cf. A305199, A326755, A326857, A326859, A326861.

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

a(n) ~ exp(-2*gamma/3) * Gamma(1/3)^2 * n! / (2 * 3^(1/6) * Pi * n^(1/3)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

A326861 E.g.f.: Product_{k>=1} (1 + x^(3k-2)/(3k-2)) / (1 - x^(3k-2)/(3k-2)).

Original entry on oeis.org

1, 2, 4, 12, 60, 360, 2160, 16560, 149040, 1386720, 14592960, 174208320, 2173897440, 29413264320, 437473872000, 6792952636800, 112213292716800, 2002551280012800, 37194983281843200, 726119227314201600, 15112608758893324800, 326665495054151193600

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

In general, if c > 0, d = 1-c and e.g.f. = Product_{k>=1} (1 + x^(c*k+d)/(c*k+d)) / (1 - x^(c*k+d)/(c*k+d)), then a(n) ~ 2 * n^(2/c) * n! / (c^(2/c) * exp(2*gamma/c) * Gamma(1 + 2/c)^2), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

Cf. A305199, A326756, A326858, A326860.

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

a(n) ~ 3^(7/3) * exp(-2*gamma/3) * Gamma(1/3)^2 * n^(2/3) * n! / (8 * Pi^2), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

A326862 E.g.f.: Product_{k>=1} (1 + x^(4k-1)/(4k-1)) / (1 - x^(4k-1)/(4k-1)).

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 160, 1440, 0, 26880, 691200, 7257600, 11827200, 395366400, 14125363200, 185119334400, 442810368000, 24049778688000, 919255538073600, 13662913904640000, 54833495408640000, 3627817738960896000, 142917996623560704000, 2442221696292618240000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A326855, A326779.

Cf. A305199, A326859, A326860, A326863.

nmax = 30; CoefficientList[Series[Product[(1+x^(4*k-1)/(4*k-1))/(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/2) * n! / (2*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A326887 E.g.f.: Product_{k>=1} (1 + (exp(x)-1)^k/k) / (1 - (exp(x)-1)^k/k).

Original entry on oeis.org

1, 2, 8, 48, 364, 3320, 35464, 433692, 5962548, 90931152, 1522657264, 27765229844, 547487475484, 11604952395816, 263091290017560, 6351255101776812, 162643987129698628, 4403250400372110656, 125649232950852714496, 3769013390615951560068, 118555772298034094231724

Offset: 0

Vaclav Kotesovec, Jul 31 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A305199, A305986, A305987.

nmax = 20; CoefficientList[Series[Product[(1+(Exp[x]-1)^k/k)/(1-(Exp[x]-1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} A305199(k)*Stirling2(n,k).

a(n) ~ n * (n+1)! / (16 * exp(2*gamma) * log(2)^(n+3)), where gamma is the Euler-Mascheroni constant A001620.

A306041 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)/(1 - x^k/k!).

Original entry on oeis.org

1, 2, 6, 26, 126, 742, 4986, 37942, 321502, 3026150, 31198206, 351179182, 4282131354, 56334933358, 795191463982, 12001157392246, 192825757504222, 3288240179785318, 59314678786251486, 1128751491248706814, 22599321692994969886, 474961934284902165190, 10454818842695667265942

Offset: 0

Ilya Gutkovskiy, Jun 17 2018

nonn

Exponential convolution of the sequences A005651 and A007837.

Cf. A005651, A007837, A305199.

a:=series(mul((1+x^k/k!)/(1-x^k/k!),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)/(1 - x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(1 + (-1)^(k + 1)) x^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (1 + (-1)^(k+1))*x^(j*k)/(k*(j!)^k)).

A326863 E.g.f.: Product_{k>=1} (1 + x^(4k-3)/(4k-3)) / (1 - x^(4k-3)/(4k-3)).

Original entry on oeis.org

1, 2, 4, 12, 48, 288, 2016, 14112, 112896, 1096704, 12063744, 135894528, 1630734336, 22157549568, 331366920192, 5107664314368, 82057393668096, 1436821272133632, 27168078863794176, 528845513033908224, 10627947138360803328, 228216184936879620096, 5219125284175176794112

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Cf. A326856, A326780.

Cf. A305199, A326859, A326861, A326862.

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

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

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

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A326859 E.g.f.: Product_{k>=1} (1 + x^(2*k-1)/(2*k-1)) / (1 - x^(2*k-1)/(2*k-1)).

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A326860 E.g.f.: Product_{k>=1} (1 + x^(3*k-1)/(3*k-1)) / (1 - x^(3*k-1)/(3*k-1)).

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A326861 E.g.f.: Product_{k>=1} (1 + x^(3*k-2)/(3*k-2)) / (1 - x^(3*k-2)/(3*k-2)).

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A326862 E.g.f.: Product_{k>=1} (1 + x^(4*k-1)/(4*k-1)) / (1 - x^(4*k-1)/(4*k-1)).

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A326887 E.g.f.: Product_{k>=1} (1 + (exp(x)-1)^k/k) / (1 - (exp(x)-1)^k/k).

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Mathematica

Formula

A306041 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)/(1 - x^k/k!).

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Maple

Mathematica

Formula

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

A326859 E.g.f.: Product_{k>=1} (1 + x^(2k-1)/(2k-1)) / (1 - x^(2k-1)/(2k-1)).

A326860 E.g.f.: Product_{k>=1} (1 + x^(3k-1)/(3k-1)) / (1 - x^(3k-1)/(3k-1)).

A326861 E.g.f.: Product_{k>=1} (1 + x^(3k-2)/(3k-2)) / (1 - x^(3k-2)/(3k-2)).

A326862 E.g.f.: Product_{k>=1} (1 + x^(4k-1)/(4k-1)) / (1 - x^(4k-1)/(4k-1)).

A326863 E.g.f.: Product_{k>=1} (1 + x^(4k-3)/(4k-3)) / (1 - x^(4k-3)/(4k-3)).