%I #9 Dec 29 2023 10:51:51
%S 1,2,9,58,469,4530,50491,634790,8861043,135750454,2262315973,
%T 40726646802,787471241647,16275700505510,358103286781293,
%U 8357593147404346,206241859929682177,5366082228239257410
%N Self-convolution square-root of A062817 (offset 2); thus g.f. A(x) satisfies: A(x)^2 = Sum_{n>=2} A062817(n)*x^n, where A062817(n) = Sum_{k=0..n} (n-k)^k*k^(n-k).
%H Vaclav Kotesovec, <a href="/A132608/b132608.txt">Table of n, a(n) for n = 1..425</a>
%F a(n) ~ exp(1) * sqrt(2*Pi/3) * n^(n + 3/2) / 2^(n+3). - _Vaclav Kotesovec_, Nov 22 2021
%e A(x) = x + 2x^2 + 9x^3 + 58x^4 + 469x^5 + 4530x^6 +...+ a(n)*x^n +...
%e A(x)^2 = x^2 + 4x^3 + 22x^4 + 152x^5 + 1251x^6 +...+ A062817(n)*x^n +...
%t nmax = 20; CoefficientList[Series[(Sum[x^(k-2) * Sum[(k-j)^j * j^(k-j), {j, 0, k}], {k, 1, 2*nmax}])^(1/2), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 22 2021 *)
%o (PARI) {a(n)=polcoeff((sum(m=2,n+1,sum(k=0,m,(m-k)^k*k^(m-k))*x^m +x*O(x^(n+1))))^(1/2),n)}
%Y Cf. A062817, A132609.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 26 2007