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A111723
Number of partitions of an n-set with an odd number of blocks of size 1.
9
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
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
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Nov 17 2005
EXTENSIONS
More terms from Robert G. Wilson v, Nov 22 2005
STATUS
approved