A385762 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^3*A''(x)).
1, 1, 2, 9, 80, 1204, 27788, 918831, 41389972, 2443323132, 183303840972, 17050267807478, 1926895029660880, 260150110806399232, 41365993162914888760, 7652990621445212758255, 1630131235132495370561820, 396129991240222795968202788, 108937459572870420021782788268
Offset: 0
Keywords
Programs
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Mathematica
terms = 19; A[] = 0; Do[A[x] = 1/(1-x*A[x]-x^3*A''[x]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
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PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+sum(k=1, 2, stirling(2, k, 1)*j^k))*v[j+1]*v[i-j])); v;
Formula
a(0) = 1; a(n) = Sum_{k=0..n-1} (1 - k + k^2) * a(k) * a(n-1-k).