OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..100
MathOverflow, asymptotic growth of a sum involving partitions, Sep 26 2021.
FORMULA
a(n) = Sum_{k=1..n} n!*(n-1)!/(n-k)!^2 * b(k) * a(n-k), where b(k) = Sum_{d|k} d^(1-2*k/d) and a(0) = 1 (after Vladeta Jovovic in A007841).
a(n) ~ 2 * n!^2. - Vaclav Kotesovec, Mar 05 2016
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2/2!^2 + 49*x^3/3!^2 + 856*x^4/4!^2 +...
where
A(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+b(n-i, min(i, n-i))*((i-1)!*binomial(n, i))^2 ))
end:
a:= n-> b(n$2):
seq(a(n), n=0..17); # Alois P. Heinz, Jul 27 2023
MATHEMATICA
b[k_] := b[k] = DivisorSum[k, #^(1-2*k/#) &]; a[0] = 1; a[n_] := a[n] = Sum[n!*(n-1)!/(n-k)!^2*b[k]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 23 2015, adapted from PARI *)
Table[n!^2 * SeriesCoefficient[Product[1/(1 - x^m/m^2), {m, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 05 2016 *)
PROG
(PARI) {a(n)=n!^2*polcoeff(prod(k=1, n, 1/(1-x^k/k^2 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Using logarithmic derivative: */
{b(k) = sumdiv(k, d, d^(1-2*k/d))}
{a(n) = if(n==0, 1, sum(k=1, n, n!*(n-1)!/(n-k)!^2 * b(k) * a(n-k)))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 01 2014
EXTENSIONS
Name clarified by Vaclav Kotesovec, Mar 05 2016
STATUS
approved