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
Keywords
Examples
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)*...).
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..100
- MathOverflow, asymptotic growth of a sum involving partitions, Sep 26 2021.
Programs
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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 *) -
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),", "))
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
Extensions
Name clarified by Vaclav Kotesovec, Mar 05 2016