OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1664
FORMULA
a(n) = [x^n] 1/(1-x)^n * 1/Product_{j=2..n} (1-x^j).
a(n) is n-th term of the Euler transform of n,1,1,1,... .
a(n) ~ c * 4^n / sqrt(n), where c = QPochhammer[-1, 1/2] / (8*sqrt(Pi) * QPochhammer[1/4, 1/4]) = 0.48841139329043831428669851139824427133317... - Vaclav Kotesovec, Sep 19 2017
Equivalently, c = 1/(4*sqrt(Pi)*QPochhammer(1/2)). - Vaclav Kotesovec, Mar 17 2024
EXAMPLE
a(2) = 4: 2, 1a1a, 1a1b, 1b1b.
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
end:
a:= n-> b(n$3):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
(numtheory[sigma](j)+k-1)*b(n-j, k), j=1..n)/n)
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=1,
combinat[numbpart](n), b(n-1, k) +b(n, k-1)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
MATHEMATICA
Table[SeriesCoefficient[1/(1-x)^(n-1) * Product[1/(1-x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 19 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 16 2017
STATUS
approved