A205799 E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2)! ).

1, 1, 1, 2, 5, 11, 32, 113, 365, 1373, 6072, 25279, 115633, 606321, 3051413, 16344785, 98402881, 576283953, 3523586227, 23840955908, 158428389359, 1085566420290, 8128568533790, 60203101002122, 455911264482697, 3734114950288571, 30413492882578846

Offset: 0

Author

Paul D. Hanna, Jan 31 2012

Keywords

• nonn

Comments

Number of set partitions of [n] whose block lengths are triangular numbers. - Alois P. Heinz, Jun 10 2018

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 11*x^5/5! + 32*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..603

Crossrefs

Cf. A193374, A205802, A305824.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      issqr(8*j+1), a(n-j)*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 10 2018

Mathematica

m = 30;
CoefficientList[Exp[Sum[x^(n(n+1)/2)/(n(n+1)/2)!, {n, 1, m}]] + O[x]^m, x]* Range[0, m-1]! (* Jean-François Alcover, Mar 05 2021 *)

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)}