A218003 Number of degree-n permutations of order a power of 3.

1, 1, 1, 3, 9, 21, 81, 351, 1233, 46089, 434241, 2359611, 27387801, 264333213, 1722161169, 16514298711, 163094452641, 1216239520401, 50883607918593, 866931703203699, 8473720481213481, 166915156382509221, 2699805625227141201, 28818706120636531023, 439756550972215638129, 6766483260087819272601, 77096822666547068590401, 406859605390184444341678251

Offset: 0

Author

Paul D. Hanna, Oct 17 2012

Keywords

• nonn

Comments

Differs from A053499 first at n=27. - Alois P. Heinz, Jan 25 2014

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 81*x^6/6! +...
where
log(A(x)) = x + x^3/3 + x^9/9 + x^27/27 + x^81/81 +...+ x^3^n/3^n +...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A005388, A053499.

Programs

Maple

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..3^j-1)*a(n-3^j), j=0..ilog[3](n))))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 25 2014

Mathematica

a[n_] := a[n] = If[n < 0, 0, If[n == 0, 1, Sum[Product[n-i, {i, 1, 3^j-1}]*a[n-3^j], {j, 0, Floor@Log[3, n]}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 28 2025, after Alois P. Heinz *)

PARI

{a(n)=n!*polcoeff(exp(sum(k=0,ceil(log(n+1)/log(3)),x^(3^k)/3^k)+x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f.: exp( Sum_{n>=0} x^(3^n)/3^n ).