%I #6 Sep 10 2024 04:08:08
%S 1,1,9,260,17215,2189997,477731884,164858203944,84745577983095,
%T 61951785517193675,62077057930391945969,82749694746019635920952,
%U 143157935882304543684640676,314805573970543375502985796300,864458294787075036217714712292600,2919280453922974335841433174057739408
%N a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^3 * a(k) * a(n-k-1).
%F G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 7 * x^2 * A(x) * A'(x) + 6 * x^3 * A(x) * A''(x) + x^4 * A(x) * A'''(x).
%t a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
%t nmax = 15; A[_] = 0; Do[A[x_] = 1 + x A[x]^2 + 7 x^2 A[x] A'[x] + 6 x^3 A[x] A''[x] + x^4 A[x] A'''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%Y Cf. A000699, A015084, A088716, A256019, A376095, A376097.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Sep 10 2024