A386264 G.f. A(x) satisfies A(x) = 1/( (1-x)^3 * (1 - x*A(x) - 2*x^2*A'(x)) ).
1, 4, 22, 190, 2239, 32431, 546832, 10414132, 219845677, 5079617326, 127292440606, 3435881715553, 99351006910147, 3063829146597493, 100385767850729656, 3483114440490487576, 127610457725933245753, 4923678697863463464970, 199592119100636938629838
Offset: 0
Keywords
Programs
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Mathematica
terms=19; A[]=1; Do[A[x]=1/( (1-x)^3 * (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]=binomial(i+2, 2)+i*sum(j=0, i-1, v[j+1]*v[i-j])); v;
Formula
a(n) = binomial(n+2,2) + n * Sum_{k=0..n-1} a(k) * a(n-1-k).
a(n) = binomial(n+2,2) + Sum_{k=0..n-1} (1 + 2*k) * a(k) * a(n-1-k).