A280241 Expansion of Product_{k>=1} 1/(1 - x^(k!!)).

1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 19, 23, 26, 31, 36, 42, 48, 56, 63, 72, 81, 91, 102, 115, 127, 142, 157, 173, 190, 210, 229, 252, 275, 300, 326, 355, 383, 416, 449, 485, 522, 563, 603, 648, 694, 743, 794, 851, 906, 968, 1031, 1097, 1166, 1241, 1315, 1398, 1481, 1569, 1660, 1758, 1855, 1962

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]

Index entries for sequences related to factorial numbers

Index entries for related partition-counting sequences

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

Cf. A006882, A064986.

Sequence in context: A020902 A008751 A029002 * A031121 A080655 A367358

Adjacent sequences: A280238 A280239 A280240 * A280242 A280243 A280244

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 29 2016

Status

approved