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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. 7
0, 1, 0, 7, 24, 201, 1560, 14743, 154896, 1813969, 23346000, 327496071, 4970498280, 81121077337, 1416223931304, 26328776843671, 519178407998880, 10821355158998433, 237677397895531296, 5485802780426178439, 132728552830731814200, 3358841601972480225001 (list; graph; refs; listen; history; text; internal format)
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 + (13/12 + (1 + exp(2))/(1 - exp(2))) / sqrt(n)). - Vaclav Kotesovec, Jan 21 2017

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-Fran├žois 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

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Last modified October 19 14:26 EDT 2021. Contains 348090 sequences. (Running on oeis4.)