A293138 E.g.f.: Product_{m>0} (1+x^m+x^(2*m)/2!).

1, 1, 3, 12, 72, 480, 3780, 35280, 372960, 4263840, 54432000, 758419200, 11436163200, 185253868800, 3214699488000, 59172265152000, 1163830187520000, 24097823253504000, 525794940582912000, 12073276215576576000, 290883846352619520000, 7318777466097377280000

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..444

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n) - n) * n^(n+1/2) / (sqrt(5) * n^(3/4)), where c = -polylog(2, -1/2 - I/2) - polylog(2, -1/2 + I/2) = 0.9669456127221570300837545... Equivalently, c = -Sum_{k>=1} (-1)^k * cos(Pi*k/4) / (k^2 * 2^(k/2-1)). - Vaclav Kotesovec, Oct 01 2017

Example

Let's consider the partitions of n where no positive integer appears more than twice. (See A000726)
For n = 5,
    partition      |                         |
--------------------------------------------------------------------
     5             -> one 5                  -> 1/(1!)       (= 1  )
   = 4 + 1         -> one 4 and one 1        -> 1/(1!*1!)    (= 1  )
   = 3 + 2         -> one 3 and one 2        -> 1/(1!*1!)    (= 1  )
   = 3 + 1 + 1     -> one 3 and two 1        -> 1/(1!*2!)    (= 1/2)
   = 2 + 2 + 1     -> two 2 and one 1        -> 1/(2!*1!)    (= 1/2)
--------------------------------------------------------------------
                                                sum             4
So a(5) = 5! * 4 = 480.
For n = 6,
    partition      |                         |
--------------------------------------------------------------------
     6             -> one 6                  -> 1/(1!)       (= 1  )
   = 5 + 1         -> one 5 and one 1        -> 1/(1!*1!)    (= 1  )
   = 4 + 2         -> one 4 and one 2        -> 1/(1!*1!)    (= 1  )
   = 4 + 1 + 1     -> one 4 and two 1        -> 1/(1!*2!)    (= 1/2)
   = 3 + 3         -> two 3                  -> 1/(2!)       (= 1/2)
   = 3 + 2 + 1     -> one 3, one 2 and one 1 -> 1/(1!*1!*1!) (= 1  )
   = 2 + 2 + 1 + 1 -> two 2 and two 1        -> 1/(2!*2!)    (= 1/4)
--------------------------------------------------------------------
                                                sum            21/4
So a(6) = 6! * 21/4 = 3780.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)/j!, j=0..min(2, n/i))))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 02 2017

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]/j!, {j, 0, Min[2, n/i]}]]];
a[n_] := n! b[n, n];
a /@ Range[0, 23] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

Crossrefs

Column k=2 of A293135.

Cf. A000726, A162891, A263401, A293141.

Sequence in context: A367117 A228386 A175836 * A319948 A020530 A337061

Adjacent sequences: A293135 A293136 A293137 * A293139 A293140 A293141

Keyword

nonn

Author

Seiichi Manyama, Oct 01 2017

Status

approved