A113235 -id:A113235 - 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

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

A113236 Number of partitions of {1,..,n} into any number of lists of size not equal to 3, where a list means an ordered subset, cf. A000262.

Original entry on oeis.org

1, 1, 3, 7, 49, 321, 2851, 24823, 256257, 2887489, 36759331, 507010791, 7597222513, 122184356737, 2106356007939, 38693238713431, 754792977928321, 15572911248409473, 338800604611562947, 7749991799652960199, 185934065196259734321, 4667877395135551746241

Offset: 0

Karol A. Penson, Oct 19 2005

nonn

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

Cf. A052845, A113235.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(1-x^2+x^3)/(1-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=3, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

Range[0, 18]!*CoefficientList[ Series[ Exp[x*(1-x^2+x^3)/(1 - x)], {x, 0, 18}], x] (* Zerinvary Lajos, Mar 23 2007 *)
a[n_] := a[n] = If[n==0, 1, Sum[If[j==3, 0, a[n-j]*Binomial[n-1, j-1]*j!], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 11 2017, after Alois P. Heinz *)

x='x+O('x^30); Vec(serlaplace(exp(x*(1-x^2+x^3)/(1-x)))) \\ G. C. Greubel, May 17 2018

E.g.f.: exp(x*(1-x^2+x^3)/(1-x)).
Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 3*k, -1, -1]/k!, {k, 0, Floor[n/3]}], n=0, 1....
a(n) ~ exp(-3/2+2*sqrt(n)-n)*n^(n-1/4)/sqrt(2). - Vaclav Kotesovec, Jun 22 2013

A113237 E.g.f.: exp(x*(1 - x^3 + x^4)/(1-x)).

Original entry on oeis.org

1, 1, 3, 13, 49, 381, 2971, 26713, 291873, 3262969, 41245651, 569262981, 8433896593, 136060620853, 2342471665899, 42987065380561, 838321137046081, 17272648375895793, 375413770580941603, 8579701021461918589, 205637099039964274161, 5158188565847339152621

Offset: 0

Karol A. Penson, Oct 19 2005

nonn

Number of partitions of {1,..,n} into any number of lists of size not equal to 4, where a list means an ordered subset, cf. A000262.

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

Cf. A000262, A052845, A097514, A113235, A113236.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=4, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}]; Table[ f[n], {n, 0, 19}]
Range[0, 19]!* CoefficientList[ Series[ Exp[x*(1 - x^3 + x^4)/(1 - x)], {x, 0, 19}], x] (* Robert G. Wilson v, Oct 21 2005 *)

Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}], n=0, 1....
Recurrence: a(n) = (2*n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) - 4*(n-3)*(n-2)*(n-1)*a(n-4) + 8*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - 4*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Jun 24 2013
a(n) ~ n^(n-1/4)*exp(-3/2+2*sqrt(n)-n)/sqrt(2) * (1 + 187/(48*sqrt(n))). - Vaclav Kotesovec, Jun 24 2013

A351823 Triangular array read by rows. T(n,k) is the number of sets of lists (as in A000262(n)) with exactly k size 2 lists, n >= 0, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 1, 2, 7, 6, 49, 12, 12, 301, 140, 60, 2281, 1470, 180, 120, 21211, 12642, 2940, 840, 220417, 127736, 41160, 3360, 1680, 2528569, 1527192, 455112, 70560, 15120, 32014801, 19837530, 5748120, 1234800, 75600, 30240, 442974511, 278142590, 83995560, 16687440, 1940400, 332640

Offset: 0

Geoffrey Critzer, Feb 20 2022

From the asymptotic estimate of A000262(n) provided by Vaclav Kotesovec we deduce that in the limit as n gets big the average number of size 2 lists is equal to 1. In other words, lim_{n->oo} Sum_{k>=1} T(n,k)*k/A000262(n) = 1. Generally for any j >= 1, the average number of size j lists equals 1 in the limit as n -> oo.

			Triangle T(n,k) begins:
      1;
      1;
      1,     2;
      7,     6;
     49,    12,   12;
    301,   140,   60;
   2281,  1470,  180, 120;
  21211, 12642, 2940, 840;
  ...

Column k=1 gives A113235.
T(n,floor(n/2)) gives A081125.
T(2n,n) gives A001813.
Cf. A000262 (row sums) A006152, A114329, A351825.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
     `if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 20 2022

nn = 7; Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Exp[ x/(1 - x) - x ^2 + y x^2], {x, 0, nn}], {x, y}]] // Grid

E.g.f.: exp(x/(1-x) - x^2 + y*x^2).

Sum_{k=0..floor(n/2)} k * T(n,k) = A351825(n). - Alois P. Heinz, Feb 24 2022

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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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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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A113236 Number of partitions of {1,..,n} into any number of lists of size not equal to 3, where a list means an ordered subset, cf. A000262.

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A113237 E.g.f.: exp(x*(1 - x^3 + x^4)/(1-x)).

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A351823 Triangular array read by rows. T(n,k) is the number of sets of lists (as in A000262(n)) with exactly k size 2 lists, n >= 0, 0 <= k <= floor(n/2).

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