A385692 E.g.f. A(x) satisfies A(x) = exp( x*A(x)*(A(x) + A(w*x) + A(w^2*x))/3 ), where w = exp(2*Pi*i/3).

1, 1, 3, 16, 189, 2256, 32167, 767313, 16423185, 385872832, 13923826371, 431494792224, 14162204393053, 685135173015801, 27831222972658029, 1174037911440510736, 71264909409165117009, 3582888868151242791360, 184756481500401258020443, 13494513883839138274687425

Offset: 0

Author

Seiichi Manyama, Jul 07 2025

Keywords

• nonn

Crossrefs

Cf. A385690.

Programs

Mathematica

terms = 20; w = Exp[2*Pi*I/3]; A[] = 0; Do[A[x] = Exp[x*A[x]*(A[x] + A[w*x] + A[w^2*x])/3 ] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]!//Simplify (* Stefano Spezia, Jul 07 2025 *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{i, j, k>=0 and i+j+3*k=n-1} (n-i) * a(i) * a(j) * a(3*k)/(i! * j! * (3*k)!).