A385763 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^4*A'''(x)).
1, 1, 2, 5, 44, 1188, 74880, 9211479, 1962123260, 665169218468, 337242780292376, 243827199998597254, 242120748323922920272, 320325994582940359050400, 550640627320172764415124000, 1204251372776149567847238889047, 3291219553094816112273747054673476
Offset: 0
Keywords
Programs
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Mathematica
terms = 17; A[] = 0; Do[A[x] = 1/(1-x*A[x]-x^4*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, 3, stirling(3, k, 1)*j^k))*v[j+1]*v[i-j])); v;
Formula
a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + 2*k - 3*k^2 + k^3) * a(k) * a(n-1-k).