A205801 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) / (n^2) ).
1, 1, 1, 1, 7, 31, 91, 211, 1681, 52417, 461161, 2427481, 10744471, 219643711, 2619643027, 18939628891, 1410692293921, 23943786881281, 263853697605841, 2237281161036337, 53316533506210471, 900164075618402911, 11265158441537890891, 112769404714319769571
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + 91*x^6/6! +... where log(A(x)) = x + x^4/4 + x^9/9 + x^16/16 + x^25/25 + x^36/36 +...
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..451
- David Harry Richman and Andrew O'Desky, Derangements and the p-adic incomplete gamma function, arXiv:2012.04615 [math.NT], 2020.
- Wikipedia, Liouville Function.
Crossrefs
Programs
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Maple
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(j), a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n)) end: seq(a(n), n=0..25); # Alois P. Heinz, May 12 2016 -
Mathematica
a[n_] := a[n] = If[n==0, 1, Sum[If[IntegerQ @ Sqrt[j], a[n-j]*(j-1)! * Binomial[n-1, j-1], 0], {j, 1, n}]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 19 2017, after Alois P. Heinz *) nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 17 2019 *) -
PARI
{a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(n+1), x^(m^2)/(m^2)+x*O(x^n))), n)} -
PARI
a(n) = if(n==0, 1, (n-1)!*sum(k=1, sqrtint(n), a(n-k^2)/(n-k^2)!)); \\ Seiichi Manyama, Apr 29 2022
Formula
The e.g.f. A(x)=1+a(1)x+a(2)x^2/2!+... is equal to the power series expansion of the product of (1-x^n)^{-lambda(n)/n} (n=1,2,...) where lambda(n) is the Liouville function A008836 (follows easily from the Lambert series of lambda(n) - see e. g., the Wikipedia link). - Mamuka Jibladze, Jan 12 2014
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor(sqrt(n))} a(n-k^2)/(n-k^2)!. - Seiichi Manyama, Apr 29 2022
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