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A218683 E.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n/n! * A(n*x).

%I A218683 #7 Jul 03 2025 08:38:40
%S A218683 1,1,6,69,1432,52065,3202176,324172597,53099867136,13888279032129,
%T A218683 5736880791449920,3710252136325373661,3729910949734728414624,
%U A218683 5792791811385586637686849,13826260704559808415109532256,50488064853691920270244556417445
%N A218683 E.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n/n! * A(n*x).
%F A218683 a(n) = Sum_{k=0..n-1} binomial(n,k) * (n-k)^n * a(k) for n>0 with a(0)=1.
%F A218683 From _Vaclav Kotesovec_, Jul 03 2025: (Start)
%F A218683 a(n) ~ c * n! * 3^(n*(n+1)/6) / 2^(n/3), where
%F A218683 c = 26660.4828462502037986697479653... if mod(n,3) = 0,
%F A218683 c = 26660.4828502439141044598280378... if mod(n,3) = 1,
%F A218683 c = 26660.4828514026721542320888687... if mod(n,3) = 2. (End)
%e A218683 E.g.f.: A(x) = 1 + x + 6*x^2/2! + 69*x^3/3! + 1432*x^4/4! + 52065*x^5/5! +...
%e A218683 where
%e A218683 A(x) = 1 + x*A(x) + 2^2*x^2*A(2*x)/2! + 3^3*x^3*A(3*x)/3! + 4^4*x^4*A(4*x)/4! +...
%e A218683 which leads to the recurrence illustrated by:
%e A218683 a(1) = 1*1^1*(1) = 1;
%e A218683 a(2) = 1*2^2*(1) + 2*1^2*(1) = 6;
%e A218683 a(3) = 1*3^3*(1) + 3*2^3*(1) + 3*1^3*(6) = 69;
%e A218683 a(4) = 1*4^4*(1) + 4*3^4*(1) + 6*2^4*(6) + 4*1^4*(69) = 1432;
%e A218683 a(5) = 1*5^5*(1) + 5*4^5*(1) + 10*3^5*(6) + 10*2^5*(69) + 5*1^5*(1432) = 52065.
%t A218683 Clear[a]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n,k] * (n-k)^n * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 03 2025 *)
%o A218683 (PARI) {a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,k^k*x^k/k!*subst(A,x,k*x)+x*O(x^n)));n!*polcoeff(A,n)}
%o A218683 for(n=0,20,print1(a(n),", "))
%o A218683 (PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, binomial(n, k)*(n-k)^n*a(k)))}
%o A218683 for(n=0,20,print1(a(n),", "))
%Y A218683 Cf. A125281, A218682.
%K A218683 nonn
%O A218683 0,3
%A A218683 _Paul D. Hanna_, Nov 05 2012