A386230 G.f. A(x) satisfies A(x) = 1/( (1-x)^3 * (1 - x*A(x) - x^2*A'(x)) ).
1, 4, 18, 114, 945, 9399, 106645, 1342028, 18409725, 272154510, 4300884555, 72225827628, 1283066570500, 24025524690426, 472822444534395, 9755834028122904, 210600429263424372, 4747647482075588598, 111583282733838959542, 2729989048854423409090, 69430953497076613542366
Offset: 0
Keywords
Programs
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Mathematica
terms = 21; A[] = 1; Do[A[x] = 1/((1-x)^3(1-x*A[x]-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]=binomial(i+2, 2)+(i+1)/2*sum(j=0, i-1, v[j+1]*v[i-j])); v;
Formula
a(n) = binomial(n+2,2) + (n+1)/2 * Sum_{k=0..n-1} a(k) * a(n-1-k).
a(n) = binomial(n+2,2) + Sum_{k=0..n-1} (1 + k) * a(k) * a(n-1-k).