A062282 -id:A062282 - cp's OEIS frontend

A111723 Number of partitions of an n-set with an odd number of blocks of size 1.

1, 0, 4, 4, 31, 86, 449, 1968, 10420, 56582, 333235, 2069772, 13606113, 94065232, 682242552, 5175100432, 40954340995, 337362555010, 2886922399649, 25616738519384, 235313456176512, 2234350827008170, 21899832049913999, 221292603495494488, 2302631998398438321

Offset: 1

Vladeta Jovovic, Nov 17 2005

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A097514, A113235, A063083, A062282, A111724, A111752, A111753.

b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
      `if`(j=1, 1-t, t))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, May 10 2016

Rest[ Range[0, 23]! CoefficientList[ Series[ Sinh[x]Exp[Exp[x] - 1 - x], {x, 0, 23}], x]] (* Robert G. Wilson v *)

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, t):
    return t if n==0 else sum(b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1) for j in range(1, n + 1))
def a(n):
    return b(n, 0)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 10 2017

E.g.f.: sinh(x)*exp(exp(x)-1-x).

More generally, e.g.f. for number of partitions of an n-set with an odd number of blocks of size k is sinh(x^k/k!)*exp(exp(x)-1-x^k/k!).

More terms from Robert G. Wilson v, Nov 22 2005

A111724 Number of partitions of an n-set with an even number of blocks of size 1.

Original entry on oeis.org

0, 2, 1, 11, 21, 117, 428, 2172, 10727, 59393, 345335, 2143825, 14038324, 96834090, 700715993, 5305041715, 41910528809, 344714251149, 2945819805408, 26107419715988, 239556359980239, 2272364911439153, 22252173805170347, 224666265799310801, 2335958333831561032

Offset: 1

Vladeta Jovovic, Nov 17 2005

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A097514, A113235, A063083, A062282, A111723, A111752, A111753.

b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
      `if`(j=1, 1-t, t))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..30);  # Alois P. Heinz, May 10 2016

Rest[ Range[0, 24]! CoefficientList[ Series[ Cosh[x]Exp[Exp[x] - 1 - x], {x, 0, 23}], x]] (* Robert G. Wilson v *)

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, t): return t if n==0 else sum(b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1) for j in range(1, n + 1))
def a(n): return b(n, 1)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 10 2017

E.g.f.: cosh(x)*exp(exp(x)-1-x).

More generally, e.g.f. for number of partitions of an n-set with an even number of blocks of size k is cosh(x^k/k!)*exp(exp(x)-1-x^k/k!).

More terms from Robert G. Wilson v, Nov 22 2005

A063083 Number of permutations of n elements with an odd number of fixed points.

Original entry on oeis.org

0, 1, 0, 4, 8, 56, 304, 2192, 17408, 156928, 1568768, 17257472, 207087616, 2692143104, 37689995264, 565349945344, 9045599092736, 153775184642048, 2767953323425792, 52591113145352192, 1051822262906519552, 22088267521037959168, 485941885462833004544

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 05 2001

nonn

Harry J. Smith, Table of n, a(n) for n=0..100

Cf. A062282.

nn = 20; d = Exp[-x]/(1 - x); Range[0, nn]! CoefficientList[Series[Sinh[x]  d, {x, 0, nn}], x] (* Geoffrey Critzer, Jan 14 2012 *)
a[n_] := -n!/2 Sum[(-2)^i/i!, {i, 1, n}]
Table[a[n], {n, 0, 20}] (* Gerry Martens , May 06 2016 *)

{ for (n=0, 100, if (n, a=n*a + (-2)^(n-1), a=0); write("b063083.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 17 2009

E.g.f.: sinh(x) * exp(-x)/(1-x). Asymptotic expression: a(n) ~ n! * (1 - 1/e^2)/2 i.e. as n goes to infinity the fraction for permutations that has an odd number of fixed points is about (1 - 1/e^2)/2 = 0.432332...
a(n) = n! - A062282(n) = n! - sum k=0 ... [n/2] sum l=0...n-2k (-1)^l * n!/((2k)! * l!)
Recurrence: a(n) = n*a(n-1)+(-2)^(n-1). - Vladeta Jovovic, Apr 11 2003
More generally, e.g.f. for number of degree-n permutations with an odd number of k-cycles is sinh(x^k/k)*exp(-x^k/k)/(1-x). - Vladeta Jovovic, Jan 31 2006
a(n) = (Gamma(n+1) - Gamma(n+1,-2)*exp(-2))/2, where Gamma(a,x) is the incomplete gamma function. - Ilya Gutkovskiy, May 06 2016

More terms from Wouter Meeussen, Aug 09 2001

A111752 Number of partitions of {1,..,n} into lists with an even number of lists of size 1, where a list means an ordered subset (cf. A000262).

Original entry on oeis.org

1, 0, 3, 6, 49, 300, 2491, 22890, 239457, 2782584, 35595091, 496577070, 7499663953, 121855323876, 2118793593099, 39245026343250, 771255810671041, 16025261292247920, 350956070419872547, 8078570913162379734, 194969375055353840241, 4922311437793379501340

Offset: 0

Vladeta Jovovic, Nov 19 2005; corrected Jun 06 2006

a(n) + A111753(n) = A000262(n). - David Wasserman, Feb 11 2009

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A113235, A063083, A062282, A111723, A111724, A111753.

b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
     `if`(j=1, 1-t, t))*binomial(n-1, j-1)*j!, j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

b[n_, t_] := b[n, t] = If[n == 0, t, Sum[b[n-j, If[j == 1, 1-t, t]] * Binomial[n-1, j-1]*j!, {j, 1, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *)

from sympy.core.cache import cacheit
from sympy import binomial, factorial as f
@cacheit
def b(n, t): return t if n==0 else sum(b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1)*f(j) for j in range(1, n + 1))
def a(n): return b(n, 1)
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 10 2017

E.g.f.: cosh(x)*exp(x^2/(1-x)). More generally, e.g.f. for number of partitions of {1, 2, ...n} into lists with an even number of lists of size k is cosh(x^k)*exp(x/(1-x)-x^k).
E.g.f.: cosh(x)*exp(x^2/(1-x)) = 1/2*Q(0); Q(k) = 1+((2*x-1)^k)/(1-x/(x+((2*x-1)^k)*(k+1)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
a(n) ~ (exp(1)+exp(-1)) * 2^(-3/2) * exp(2*sqrt(n)-n-3/2) * n^(n-1/4) * (1 + (2/(1 + exp(2)) - 5/48)/sqrt(n)). - Vaclav Kotesovec, Jan 21 2017, extended Dec 01 2021

More terms from David Wasserman, Feb 11 2009

a(0)=1 prepended by Alois P. Heinz, May 10 2016

A111753 Number of partitions of {1,..,n} into lists with an odd number of lists of size 1, where a list means an ordered subset, cf. A000262.

Original entry on oeis.org

0, 1, 0, 7, 24, 201, 1560, 14743, 154896, 1813969, 23346000, 327496071, 4970498280, 81121077337, 1416223931304, 26328776843671, 519178407998880, 10821355158998433, 237677397895531296, 5485802780426178439, 132728552830731814200, 3358841601972480225001

Offset: 0

Vladeta Jovovic, Nov 19 2005; corrected Jun 06 2006

a(n) + A111752(n) = A000262(n). - David Wasserman, Feb 11 2009

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A113235, A063083, A062282, A111723, A111724, A111752.

b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
     `if`(j=1, 1-t, t))*binomial(n-1, j-1)*j!, j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

b[n_, t_] := b[n, t] = If[n==0, t, Sum[b[n-j, If[j==1, 1-t, t]]*Binomial[ n-1, j-1]*j!, {j, 1, n}]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)

from sympy.core.cache import cacheit
from sympy import binomial, factorial as f
@cacheit
def b(n, t): return t if n==0 else sum([b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1)*f(j) for j in range(1, n + 1)])
def a(n): return b(n, 0)
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 10 2017

E.g.f.: sinh(x)*exp(x^2/(1-x)). More generally, e.g.f. for number of partitions of {1, 2, ...n} into lists with an odd number of lists of size k is sinh(x^k)*exp(x/(1-x)-x^k).
E.g.f.: sinh(x)*exp(x^2/(1-x))=1/2*Q(0);   Q(k)=1-((2x-1)^k)/( 1-x/(x-((2x-1)^k)*(k+1)*(1-x)/Q(k+1)));  (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
a(n) ~ (exp(1)-exp(-1)) * 2^(-3/2) * exp(2*sqrt(n)-n-3/2) * n^(n-1/4) * (1 + (43/48 - coth(1))/sqrt(n)). - Vaclav Kotesovec, Dec 01 2021

More terms from David Wasserman, Feb 11 2009

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A088336 Number of permutations in the symmetric group S_n that have even number of transpositions in their cycle decomposition.

Original entry on oeis.org

1, 1, 1, 3, 18, 90, 480, 3360, 27720, 249480, 2479680, 27276480, 327650400, 4259455200, 59623724160, 894355862400, 14309953257600, 243269205379200, 4378836875212800, 83197900629043200, 1663958347802150400, 34943125303845158400, 768748742605299456000

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 07 2003

nonn

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

Cf. A000142, A062282, A088506.

mx = 21; Rest[ Range[0, mx]! CoefficientList[ Series[(Exp[-x^2] + 1)/(1 - x)/2, {x, 0, mx}], x]] (* Robert G. Wilson v, May 04 2013 *)

x='x+O('x^50); Vec(serlaplace((exp(-x^2)+1)/(1-x)/2)) \\ G. C. Greubel, Aug 20 2017

E.g.f.: (exp(-x^2)+1)/(1-x)/2. - Vladeta Jovovic, Nov 09 2003

a(n) ~ n! * (1+exp(-1))/2. - Vaclav Kotesovec, Oct 08 2013

More terms from Robert G. Wilson v, May 04 2013

a(0) = 1 prepended by Alois P. Heinz, Jun 14 2015

cp's OEIS Frontend

A111723 Number of partitions of an n-set with an odd number of blocks of size 1.

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A111724 Number of partitions of an n-set with an even number of blocks of size 1.

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A063083 Number of permutations of n elements with an odd number of fixed points.

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A111752 Number of partitions of {1,..,n} into lists with an even number of lists of size 1, where a list means an ordered subset (cf. A000262).

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A111753 Number of partitions of {1,..,n} into lists with an odd number of lists of size 1, where a list means an ordered subset, cf. A000262.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A088336 Number of permutations in the symmetric group S_n that have even number of transpositions in their cycle decomposition.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions