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A205801 E.g.f.: exp( Sum_{n>=1} x^(n^2) / (n^2) ). 15
1, 1, 1, 1, 7, 31, 91, 211, 1681, 52417, 461161, 2427481, 10744471, 219643711, 2619643027, 18939628891, 1410692293921, 23943786881281, 263853697605841, 2237281161036337, 53316533506210471, 900164075618402911, 11265158441537890891, 112769404714319769571 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Number of permutations of [n] whose cycle lengths are squares. - Alois P. Heinz, May 12 2016

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..451

Wikipedia, Liouville Function.

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

EXAMPLE

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 +...

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-Fran├žois 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 *)

PROG

(PARI) {a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(n+1), x^(m^2)/(m^2)+x*O(x^n))), n)}

CROSSREFS

Cf. A000290, A193374, A205800, A205802, A273001, A273997, A308397, A317129, A329945.

Sequence in context: A118934 A118935 A226838 * A193437 A199921 A192596

Adjacent sequences:  A205798 A205799 A205800 * A205802 A205803 A205804

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 31 2012

STATUS

approved

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Last modified February 28 23:19 EST 2020. Contains 332353 sequences. (Running on oeis4.)