A386229 G.f. A(x) satisfies A(x) = 1/( (1-x)^2 * (1 - x*A(x) - x^2*A'(x)) ).
1, 3, 12, 70, 535, 4908, 51478, 600584, 7662285, 105684465, 1563183259, 24645719004, 412279514088, 7290426692472, 135862518564330, 2661378323466016, 54675576786754501, 1175673956931922257, 26411686616265112230, 618863341216409971750, 15101129008183181824938
Offset: 0
Keywords
Programs
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Mathematica
terms = 21; A[] = 1; Do[A[x] = 1/((1-x)^2(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]=(i+1)*(1+sum(j=0, i-1, v[j+1]*v[i-j])/2)); v;
Formula
a(n) = (n+1) * (1 + 1/2 * Sum_{k=0..n-1} a(k) * a(n-1-k)).
a(n) = n + 1 + Sum_{k=0..n-1} (1 + k) * a(k) * a(n-1-k).