A385543 G.f. A(x) satisfies A(x) = Sum_{k>=0} x^k * A(k^2*x).
1, 1, 2, 7, 49, 676, 18861, 1062533, 121557594, 28281916427, 13399862563765, 12949857822909156, 25549330363139585961, 103025771800413460066681, 849971455496325163128172498, 14359775106466928789344919850719, 497276944869002836686738999984515113
Offset: 0
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = Sum[(n-k)^(2*k) * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 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, (i-j)^(2*j)*v[j+1])); v;
Formula
a(0) = 1; a(n) = Sum_{k=0..n-1} (n-k)^(2*k) * a(k).
From Vaclav Kotesovec, Jul 03 2025: (Start)
a(n) ~ c * n! * 3^(n*(n-4)/3) / 2^(n/3), where
c = 438919.4178887847632978930903514036169636302175176... if mod(n,3) = 0,
c = 438919.4215235929223401081041169940935227575106084... if mod(n,3) = 1,
c = 438919.4025215529290127441106624079221416448856280... if mod(n,3) = 2. (End)