A193374 E.g.f.: A(x) = exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2) ).

1, 1, 1, 3, 9, 21, 201, 1191, 4593, 36009, 620721, 5297931, 40360761, 474989373, 4345942329, 122776895151, 2118941145441, 21344580276561, 303071564084193, 4476037678611219, 59935820004483561, 3838519441659950181, 78361805638079449641, 949279542954821272503

Offset: 0

Author

Paul D. Hanna, Jul 24 2011

Keywords

• nonn

Comments

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

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 201*x^6/6! +...
where
log(A(x)) = x + x^3/3 + x^6/6 + x^10/10 + x^15/15 + x^21/21 +...
		

Links

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

Crossrefs

Cf. A000217, A193375, A273001, A305824, A317130.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(8*j+1),
       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[8*j + 1], a[n - j]*(j - 1)!*Binomial[n - 1, j - 1], 0], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

PARI

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