A385527 E.g.f. A(x) satisfies A(x) = exp(x*A(4*x)).

1, 1, 9, 457, 118961, 152894961, 940318147705, 26967408304580857, 3534888068831469959649, 2084993641133372935803249505, 5465706581663919414225671125834601, 63043356313898446097762231466174924913065, 3173076775252515207774429654590479617164788572049

Offset: 0

Author

Seiichi Manyama, Jul 02 2025

Keywords

• nonn

Crossrefs

Cf. A000272, A096538, A141369, A385526, A385528, A385529, A385530.

Cf. A015085.

Programs

Mathematica

nmax = 15; A[] = 1; Do[A[x] = E^(x*A[4*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 A385527(n)
  A(4, n)
end

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * 4^k * binomial(n-1,k) * a(k) * a(n-1-k).

a(n) ~ c * n! * 2^(n*(n-1)), where c = 1.216702003338638031273833889488221691367428313263423339843... - Vaclav Kotesovec, Jul 02 2025