A386534 E.g.f. A(x) satisfies A(x) = exp(x * A(x) + x^5/120 * A''''(x)).
1, 1, 3, 16, 125, 1421, 26833, 968626, 70638465, 10215072856, 2782227253373, 1347216023489436, 1099522113403916545, 1443781044602756539876, 2930977624516859360997387, 8889808786962394898290294048, 39115513670641030174644662148305, 243377943140592361750259305827057888
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Keywords
Programs
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Mathematica
terms = 18; A[] = 1; Do[A[x] = Exp[x*A[x]+x^5*A''''[x]/120] + 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, 4, stirling(4, k, 1)*j^k)/120)*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 + (-6*k + 11*k^2 - 6*k^3 + k^4)/120) * binomial(n-1,k) * a(k) * a(n-1-k).