A385921 E.g.f. A(x) satisfies A(x) = exp(x*A(x) + x^4*A'''(x)).
1, 1, 3, 16, 509, 66216, 24639367, 21043463344, 35690424280569, 108571039785256960, 549371080081204026731, 4363111116508031602712064, 51938511093491129409954627637, 892615592639462586040781503568896, 21469194967164193484102627607895188975, 703974996795045871424921458192403079479296
Offset: 0
Keywords
Programs
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Mathematica
terms = 16; A[] = 1; Do[A[x] = Exp[x*A[x]+x^4*A'''[x]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 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+j)*(1+sum(k=1, 3, stirling(3, k, 1)*j^k))*binomial(i-1, j)*v[j+1]*v[i-j])); v;
Formula
a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + 2*k - 3*k^2 + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).