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

1,3

Links

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

Formula

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!).

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, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, May 10 2016

Mathematica

Rest[ Range[0, 23]! CoefficientList[ Series[ Sinh[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, 0)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 10 2017

Crossrefs

Cf. A097514, A113235, A063083, A062282, A111724, A111752, A111753.

Sequence in context: A264586 A348635 A307499 * A337447 A239601 A196131

Adjacent sequences: A111720 A111721 A111722 * A111724 A111725 A111726

Keyword

easy,nonn

Author

Vladeta Jovovic, Nov 17 2005

Extensions

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

Status

approved