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

%I A205799 #14 Mar 05 2021 07:40:22
%S A205799 1,1,1,2,5,11,32,113,365,1373,6072,25279,115633,606321,3051413,
%T A205799 16344785,98402881,576283953,3523586227,23840955908,158428389359,
%U A205799 1085566420290,8128568533790,60203101002122,455911264482697,3734114950288571,30413492882578846
%N A205799 E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2)! ).
%C A205799 Number of set partitions of [n] whose block lengths are triangular numbers. - _Alois P. Heinz_, Jun 10 2018
%H A205799 Alois P. Heinz, <a href="/A205799/b205799.txt">Table of n, a(n) for n = 0..603</a>
%e A205799 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! +...
%e A205799 where
%e A205799 log(A(x)) = x + x^3/3! + x^6/6! + x^10/10! + x^15/15! + x^21/21! +...
%p A205799 a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
%p A205799       issqr(8*j+1), a(n-j)*binomial(n-1, j-1), 0), j=1..n))
%p A205799     end:
%p A205799 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 10 2018
%t A205799 m = 30;
%t A205799 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 *)
%o A205799 (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)}
%Y A205799 Cf. A193374, A205802, A305824.
%K A205799 nonn
%O A205799 0,4
%A A205799 _Paul D. Hanna_, Jan 31 2012