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A351798 a(0) = 1; a(n) = (1/2) * Sum_{k=0..n-1} (k+1) * (k+2) * a(k) * a(n-k-1).

%I A351798 #8 Aug 19 2022 04:53:46
%S A351798 1,1,4,31,377,6531,152452,4619130,176631345,8334329638,476245005316,
%T A351798 32437793281489,2597918907028430,241796318654003869,
%U A351798 25886976434072903664,3159556047500264255868,436160347706069120482893,67621917400663695356651589,11700923494462411106797164208
%N A351798 a(0) = 1; a(n) = (1/2) * Sum_{k=0..n-1} (k+1) * (k+2) * a(k) * a(n-k-1).
%F A351798 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 2 * x^2 * A(x) * A'(x) + x^3 * A(x) * A''(x) / 2.
%p A351798 A351798 := proc(n)
%p A351798     option remember;
%p A351798     if n = 0 then
%p A351798         1;
%p A351798     else
%p A351798         add((1+k)*(2+k)*procname(k)*procname(n-k-1),k=0..n-1) ;
%p A351798         %/2 ;
%p A351798     end if;
%p A351798 end proc:
%p A351798 seq(A351798(n),n=0..30) ; # _R. J. Mathar_, Aug 19 2022
%t A351798 a[0] = 1; a[n_] := a[n] = (1/2) Sum[(k + 1) (k + 2) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
%t A351798 nmax = 18; A[_] = 0; Do[A[x_] = 1 + x A[x]^2 + 2 x^2 A[x] A'[x] + x^3 A[x] A''[x]/2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%Y A351798 Cf. A000108, A088716.
%K A351798 nonn
%O A351798 0,3
%A A351798 _Ilya Gutkovskiy_, Feb 19 2022