A329945 -id:A329945 - cp's OEIS frontend

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

Paul D. Hanna, Jan 31 2012

nonn

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

			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 +...

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.

Cf. A000290, A193374, A205800, A205802, A273001, A273997, A308397, A317129, A329945, A353184.

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

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 *)

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

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

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

A329944 Number of permutations of [n] whose cycle lengths avoid primes.

Original entry on oeis.org

1, 1, 1, 1, 7, 31, 211, 1051, 10081, 107857, 1227241, 8969401, 108817831, 1173362191, 19426473067, 320062090531, 5692838161921, 70426164947041, 1346222143950481, 21952313047471537, 493701484264143751, 10971915198235355071, 266542798822750395331

Offset: 0

Alois P. Heinz, Nov 24 2019

nonn

			a(4) = 7: (1)(2)(3)(4), (1234), (1243), (1324), (1342), (1423), (1432).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000040, A218002, A329945, A364450.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(isprime(j), 0,
      a(n-j)*binomial(n-1, j-1)*(j-1)!), j=1..n))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n == 0, 1, Sum[If[PrimeQ[j], 0,
     a[n-j] Binomial[n-1, j-1] (j-1)!], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 31 2021, after Alois P. Heinz *)

a(n) mod 2 = 1.

a(n) mod 10 = period 5: repeat [1,1,1,1,7], g.f.: (7*x^4+x^3+x^2+x+1)/(1-x^5).

cp's OEIS Frontend

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

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