A385526 E.g.f. A(x) satisfies A(x) = exp(x*A(3*x)).

1, 1, 7, 208, 23365, 9588976, 14040296659, 71747056999360, 1255862559932597257, 74168744207577385109248, 14599375893944236344767578111, 9483024632344097320792984610415616, 20158786175666520486280070249843236771213, 139271933359690469686747131442731382830399594496

Offset: 0

Author

Seiichi Manyama, Jul 02 2025

Keywords

• nonn

Crossrefs

Cf. A000272, A096538, A141369, A385527, A385528, A385529, A385530.

Cf. A015084.

Programs

Mathematica

nmax = 15; A[] = 1; Do[A[x] = E^(x*A[3*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 02 2025 *)

Ruby

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385526(n)
  A(3, n)
end

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * 3^k * binomial(n-1,k) * a(k) * a(n-1-k).

a(n) ~ c * n! * 3^(n*(n-1)/2), where c = 1.361839192264541770366149558100099215697354561... - Vaclav Kotesovec, Jul 02 2025