A385101 E.g.f. A(x) satisfies A(x) = exp(x * A(x) + x^4/24 * A'''(x)).
1, 1, 3, 16, 141, 2161, 59842, 2979509, 258264379, 37321303420, 8597483041421, 3028595626839564, 1572449537786394577, 1165432782899826271026, 1199378312656505145280950, 1673258190849282722438631406, 3099020844849243071430739707913, 7481267275389054589164201426886656
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Keywords
Programs
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Mathematica
terms = 18; A[] = 1; Do[A[x] = Exp[x*A[x]+x^4*A'''[x]/24] + 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)/24)*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)/24) * binomial(n-1,k) * a(k) * a(n-1-k).