OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 501 terms from Seiichi Manyama)
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
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)
= 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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 02 2017
STATUS
approved