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.

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

Offset

0,4

Comments

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

Links

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

Formula

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

Maple

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

Mathematica

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 *)

Prog

(Python)
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

Crossrefs

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

Sequence in context: A009646 A293489 A009650 * A198055 A012761 A209483

Adjacent sequences: A111750 A111751 A111752 * A111754 A111755 A111756

Keyword

easy,nonn

Author

Vladeta Jovovic, Nov 19 2005; corrected Jun 06 2006

Extensions

More terms from David Wasserman, Feb 11 2009

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

Status

approved