OFFSET
0,3
COMMENTS
Number of partitions of n into double factorials parts (0!! not allowed).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
FORMULA
G.f.: Product_{k>=1} 1/(1 - x^(k!!)).
EXAMPLE
a(5) = 5 because we have [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].
MATHEMATICA
CoefficientList[Series[Product[1/(1 - x^k!!), {k, 1, 10}], {x, 0, 66}], x]
PROG
(PARI) doublefactorial(n) = prod(j=0, (n-1)\2, n - 2*j );
my(x='x+O('x^70)); Vec( prod(k=1, 10, 1/(1-x^doublefactorial(k))) ) \\ G. C. Greubel, Aug 07 2019
(Magma)
DoubleFactorial:=func< n | (&*[n..2 by -2]) >;
R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (&*[1/(1-x^DoubleFactorial(k)) :k in [1..10]]) )); // G. C. Greubel, Aug 07 2019
(Sage)
from sympy import factorial2
( product(1/(1-x^factorial2(k)) for k in (1..10)) ).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Aug 07 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 29 2016
STATUS
approved