A386266 G.f. A(x) satisfies A(x) = 1/( (1+x) * (1 - x*A(x) - 2*x^2*A'(x)) ).
1, 0, 1, 5, 41, 414, 5029, 71154, 1148369, 20813552, 418536501, 9247315593, 222692887441, 5805921628194, 162930550724369, 4897006907712089, 156949282887329569, 5343387198219575688, 192583054168461798805, 7325489802429780722647, 293278021624200874980161
Offset: 0
Keywords
Programs
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Mathematica
terms=21; A[]=1; Do[A[x]=1/( (1+x) * (1 - x*A[x] - 2*x^2*A'[x]) )+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 17 2025 *)
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PARI
a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+i*sum(j=0, i-1, v[j+1]*v[i-j])); v;
Formula
a(n) = (-1)^n + n * Sum_{k=0..n-1} a(k) * a(n-1-k).
a(n) = (-1)^n + Sum_{k=0..n-1} (1 + 2*k) * a(k) * a(n-1-k).