cp's OEIS Frontend

A205801 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) / (n^2) ).

%I A205801 #56 May 01 2022 06:02:58
%S A205801 1,1,1,1,7,31,91,211,1681,52417,461161,2427481,10744471,219643711,
%T A205801 2619643027,18939628891,1410692293921,23943786881281,263853697605841,
%U A205801 2237281161036337,53316533506210471,900164075618402911,11265158441537890891,112769404714319769571
%N A205801 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) / (n^2) ).
%C A205801 Number of permutations of [n] whose cycle lengths are squares. - _Alois P. Heinz_, May 12 2016
%H A205801 Alois P. Heinz, <a href="/A205801/b205801.txt">Table of n, a(n) for n = 0..451</a>
%H A205801 David Harry Richman and Andrew O'Desky, <a href="https://arxiv.org/abs/2012.04615">Derangements and the p-adic incomplete gamma function</a>, arXiv:2012.04615 [math.NT], 2020.
%H A205801 Wikipedia, <a href="http://en.wikipedia.org/wiki/Liouville_function">Liouville Function</a>.
%F A205801 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
%F A205801 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
%e A205801 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! +...
%e A205801 where
%e A205801 log(A(x)) = x + x^4/4 + x^9/9 + x^16/16 + x^25/25 + x^36/36 +...
%p A205801 a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(j),
%p A205801        a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
%p A205801     end:
%p A205801 seq(a(n), n=0..25);  # _Alois P. Heinz_, May 12 2016
%t A205801 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_ *)
%t A205801 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 *)
%o A205801 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(n+1), x^(m^2)/(m^2)+x*O(x^n))), n)}
%o A205801 (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
%Y A205801 Cf. A000290, A193374, A205800, A205802, A273001, A273997, A308397, A317129, A329945, A353184.
%K A205801 nonn
%O A205801 0,5
%A A205801 _Paul D. Hanna_, Jan 31 2012