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A125281 E.g.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x)/n!.

%I A125281 #20 Jul 03 2025 05:51:53
%S A125281 1,1,3,16,149,2316,59047,2429554,159549945,16557985432,2693862309131,
%T A125281 682199144788734,267277518618047797,161130714885281760100,
%U A125281 148762112860064623199295,209444428223095096806228346,447998198975235291015396393713,1450973400598977755884988875863216
%N A125281 E.g.f. satisfies: A(x) = Sum{n>=0} x^n * A(n*x)/n!.
%F A125281 a(n) = Sum_{k=0..n-1} C(n,k)*(n-k)^k * a(k) for n>0 with a(0)=1.
%e A125281 A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2316*x^5/5! +...
%e A125281 where
%e A125281 A(x) = 1 + x*A(x) + x^2*A(2*x)/2! + x^3*A(3*x)/3! + x^4*A(4*x)/4! + x^5*A(5*x)/5! +...
%e A125281 which leads to the recurrence illustrated by:
%e A125281 a(3) = 1*3^0*(1) + 3*2^1*(1) + 3*1^2*(3) = 16;
%e A125281 a(4) = 1*4^0*(1) + 4*3^1*(1) + 6*2^2*(3) + 4*1^3*(16) = 149;
%e A125281 a(5) = 1*5^0*(1) + 5*4^1*(1) + 10*3^2*(3) + 10*2^3*(16) + 5*1^4*(149) = 2316.
%t A125281 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n,k] * (n-k)^k * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 03 2025 *)
%o A125281 (PARI) {a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)*(n-k)^k*a(k)))}
%o A125281 (PARI) {a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,x^k/k!*subst(A,x,k*x)+x*O(x^n)));n!*polcoeff(A,n)}
%o A125281 for(n=0,20,print1(a(n),", "))
%Y A125281 Cf. A230323, A125282, A218683.
%K A125281 nonn
%O A125281 0,3
%A A125281 _Paul D. Hanna_, Nov 29 2006, Sep 22 2007