A385621 E.g.f. A(x) satisfies A(x) = exp( x*(A(x) + A(3*x))/2 ).
1, 1, 5, 88, 5301, 1115376, 823422553, 2109689685664, 18480955532693321, 545890697425512822016, 53732023859661557312932461, 17451409191554766719804179944960, 18549102840350232594174973144785505021, 64075761168350693656591055399719635872352256, 715103564008507527975398262955305194708318040264449
Offset: 0
Keywords
Programs
-
Mathematica
terms = 15; A[] = 1; Do[A[x] =Exp[ x*(A[x] + A[3*x])/2 ]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 05 2025 *)
-
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (j+1)*(3^j+1)*binomial(i-1, j)*v[j+1]*v[i-j])/2); v;
Formula
a(0) = 1; a(n) = (1/2) * Sum_{k=0..n-1} (k+1) * (3^k+1) * binomial(n-1,k) * a(k) * a(n-1-k).