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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). 7
1, 0, 3, 6, 49, 300, 2491, 22890, 239457, 2782584, 35595091, 496577070, 7499663953, 121855323876, 2118793593099, 39245026343250, 771255810671041, 16025261292247920, 350956070419872547, 8078570913162379734, 194969375055353840241, 4922311437793379501340 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) + A111753(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.: 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+((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). - 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, 1):

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, 1]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Jan 21 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 xrange(1, n + 1)])

def a(n): return b(n, 1)

print map(a, xrange(51)) # Indranil Ghosh, Aug 10 2017

CROSSREFS

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

Sequence in context: A279706 A032322 A203765 * A102931 A056447 A056437

Adjacent sequences:  A111749 A111750 A111751 * A111753 A111754 A111755

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Nov 19 2005; corrected Jun 06 2006

EXTENSIONS

More terms from David Wasserman, Feb 11 2009

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

STATUS

approved

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Last modified May 19 20:41 EDT 2019. Contains 323410 sequences. (Running on oeis4.)