A386533 E.g.f. A(x) satisfies A(x) = exp(x * A(x) + x^3/6 * A''(x)).
1, 1, 3, 19, 225, 4576, 149517, 7448134, 542269961, 55702422400, 7832607617351, 1468762340728464, 359026336711386577, 112153290859090469184, 44001791667365123420025, 21354097196759712722857776, 12647439446531876144344860113, 9033421564454672567830839315456
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Keywords
Programs
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Mathematica
terms = 18; A[] = 1; Do[A[x] = Exp[x*A[x]+ x^3A''[x]/6] + 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, 2, stirling(2, k, 1)*j^k)/6)*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 + (-k + k^2)/6) * binomial(n-1,k) * a(k) * a(n-1-k).