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A386505 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k^2 * binomial(n-1,k) * a(k) * a(n-1-k).

%I A386505 #13 Aug 02 2025 06:57:35
%S A386505 1,1,3,43,1717,146261,22851301,5923208845,2370243182889,
%T A386505 1386889039102537,1137386506152214441,1263728857603292729441,
%U A386505 1850186029852575829090909,3487711314718246830637945549,8300937715895750334611432889933,24529666348754849148034163067487381
%N A386505 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k^2 * binomial(n-1,k) * a(k) * a(n-1-k).
%F A386505 E.g.f. A(x) satisfies A(x) = exp( x + x^2 * (d/dx A(x)) + x^3 * (d^2/dx^2 A(x)) ).
%t A386505 A386505[0] = 1;
%t A386505 A386505[n_] := A386505[n] = If[n==0,
%t A386505             1,
%t A386505             A386505[n-1]+ Sum[(1+k)*k^2*Binomial[n-1,k]*A386505[k]*A386505[n-1-k] ,{k,0,n-1} ]
%t A386505         ] ;
%t A386505 Do [ Print[A386505[n]],{n,0,20}] (* _R. J. Mathar_, Aug 02 2025 *)
%o A386505 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, (1+j)*j^2*binomial(i-1, j)*v[j+1]*v[i-j])); v;
%Y A386505 Cf. A143925, A386506, A386507, A386508, A386509.
%Y A386505 Cf. A385939, A386443.
%K A386505 nonn
%O A386505 0,3
%A A386505 _Seiichi Manyama_, Jul 24 2025