%I #6 Jan 24 2020 20:57:15

%S 1,1,5,32,280,3280,49480,927560,21037640,566134160,17803754560,

%T 646052181520,26757321804880,1252934215973600,65791336312915520,

%U 3846554938702140320,248841434876849499040,17713758333248102781760,1380631354206969100115200

%N E.g.f. A(x) satisfies: d/dx A(x) = 1 + (1/(1 - x)) * A(x/(1 - x)).

%F a(1) = 1; a(n+1) = Sum_{k=0..n-1} binomial(n,k)^2 * k! * a(n-k).

%t terms = 20; A[_] = 0; Do[A[x_] = Normal[Integrate[1 + 1/(1 - x) A[x/(1 - x) + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]! // Rest

%t a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^2 k! a[n - k - 1], {k, 0, n - 2}]; Table[a[n], {n, 1, 20}]

%Y Cf. A001063, A110083, A132228, A331661.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Jan 23 2020