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

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

1,2

Links

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

Formula

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!).

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=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..30);  # Alois P. Heinz, May 10 2016

Mathematica

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

Prog

(Python)
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

Crossrefs

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

Sequence in context: A300455 A270264 A305877 * A184299 A080371 A151337

Adjacent sequences: A111721 A111722 A111723 * A111725 A111726 A111727

Keyword

easy,nonn

Author

Vladeta Jovovic, Nov 17 2005

Extensions

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

Status

approved