A249588 G.f.: Product_{n>=1} 1/(1 - x^n/n^2) = Sum_{n>=0} a(n)*x^n/n!^2.

1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, 229357803456, 23289083584704, 2851295406197184, 414855423241758720, 70695451937596732416, 13958230719814052097024, 3159974451734082088897536, 813380358295803762813321216, 236172126115504055456155975680

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

Cf. A249590, A249607, A007841, A249593, A249592, A269791, A269793, A269794.

Sequence in context: A305114 A001819 A064618 * A348901 A363480 A193199

Adjacent sequences: A249585 A249586 A249587 * A249589 A249590 A249591

Keyword

nonn

Author

Paul D. Hanna, Nov 01 2014

Extensions

Name clarified by Vaclav Kotesovec, Mar 05 2016

Status

approved