A386239 G.f. A(x) satisfies A(x) = 1/( (1+x) * (1 - x^2*A'(x)) ).
1, -1, 0, 0, 1, 3, 11, 47, 253, 1651, 12610, 109744, 1069355, 11520785, 135906642, 1741702304, 24089599321, 357592702647, 5669840845462, 95623921546478, 1709172658222253, 32271612381443479, 641820925099092985, 13410242452064469153, 293676423537521878381
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Keywords
Programs
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Mathematica
terms = 25; A[] = 1; Do[A[x] = 1/((1+x)(1-x^2*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 16 2025 *)
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PARI
a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+(i-1)/2*sum(j=0, i-1, v[j+1]*v[i-j])); v;
Formula
a(n) = (-1)^n + (n-1)/2 * Sum_{k=0..n-1} a(k) * a(n-1-k).
a(n) = (-1)^n + Sum_{k=0..n-1} k * a(k) * a(n-1-k).