A293204 G.f.: Product_{m>0} (1+x^m+2!*x^(2*m)).

1, 1, 3, 2, 6, 7, 12, 13, 22, 26, 42, 46, 73, 80, 116, 139, 194, 226, 306, 358, 482, 558, 735, 856, 1108, 1300, 1657, 1926, 2426, 2834, 3530, 4110, 5082, 5898, 7234, 8409, 10216, 11860, 14304, 16568, 19891, 22990, 27470, 31670, 37630, 43382, 51274, 58982, 69450

Offset: 0

Author

Seiichi Manyama, Oct 02 2017

Keywords

• nonn

Examples

			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)
   = 4 + 1         -> one 4 and one 1        -> 1!*1!    (= 1)
   = 3 + 2         -> one 3 and one 2        -> 1!*1!    (= 1)
   = 3 + 1 + 1     -> one 3 and two 1        -> 1!*2!    (= 2)
   = 2 + 2 + 1     -> two 2 and one 1        -> 2!*1!    (= 2)
--------------------------------------------------------------
                                                a(5)      = 7.
For n = 6,
    partition      |                         |
--------------------------------------------------------------
     6             -> one 6                  -> 1!       (= 1)
   = 5 + 1         -> one 5 and one 1        -> 1!*1!    (= 1)
   = 4 + 2         -> one 4 and one 2        -> 1!*1!    (= 1)
   = 4 + 1 + 1     -> one 4 and two 1        -> 1!*2!    (= 2)
   = 3 + 3         -> two 3                  -> 2!       (= 2)
   = 3 + 2 + 1     -> one 3, one 2 and one 1 -> 1!*1!*1! (= 1)
   = 2 + 2 + 1 + 1 -> two 2 and two 1        -> 2!*2!    (= 4)
--------------------------------------------------------------
                                                a(6)      = 12.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 501 terms from Seiichi Manyama)

Crossrefs

Column k=2 of A293202.

Cf. A000726, A263401, A293138, A293182.

Cf. A293072.

Programs

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-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 02 2017

Mathematica

nmax = 100; CoefficientList[Series[Product[1 + x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 02 2017 *)

Formula

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4 * sqrt(Pi) * n^(3/4)), where c = Pi^2/3 - arctan(sqrt(7))^2 + log(2)^2/4 + polylog(2, -1/4 - I*sqrt(7)/4) + polylog(2, -1/4 + I*sqrt(7)/4) = 1.323865936864425754643630663383779192757247984691212163137... - Vaclav Kotesovec, Oct 02 2017

Equivalently, c = -polylog(2, -1/2 + I*sqrt(7)/2) - polylog(2, -1/2 - I*sqrt(7)/2). - Vaclav Kotesovec, Oct 05 2017