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%I #18 Jul 22 2018 11:23:20
%S 1,1,1,3,9,21,201,1191,4593,36009,620721,5297931,40360761,474989373,
%T 4345942329,122776895151,2118941145441,21344580276561,303071564084193,
%U 4476037678611219,59935820004483561,3838519441659950181,78361805638079449641,949279542954821272503
%N E.g.f.: A(x) = exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2) ).
%C Number of permutations of [n] whose cycle lengths are triangular numbers. - _Alois P. Heinz_, May 12 2016
%H Alois P. Heinz, <a href="/A193374/b193374.txt">Table of n, a(n) for n = 0..451</a>
%e 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! +...
%e where
%e log(A(x)) = x + x^3/3 + x^6/6 + x^10/10 + x^15/15 + x^21/21 +...
%p a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(8*j+1),
%p a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 12 2016
%t 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}]];
%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 05 2018, after _Alois P. Heinz_ *)
%o (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 Cf. A000217, A193375, A273001, A305824, A317130.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jul 24 2011
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