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A168408 E.g.f.: Sum_{n>=0} (exp(3^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).

%I A168408 #9 Jul 02 2025 11:24:07
%S A168408 1,3,90,21897,46281375,882516062106,153201395082609531,
%T A168408 241604140428719375972139,3448358784659838019970862469260,
%U A168408 444238039567848645977924129826080612043
%N A168408 E.g.f.: Sum_{n>=0} (exp(3^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).
%F A168408 a(n) = exp(-3^n)*Sum_{k>=0} (3^k*k)^n/k!.
%F A168408 a(n) = [x^n/n! ] Bell(x)^(3^n) where Bell(x) = exp(exp(x) - 1) is the e.g.f. of the Bell numbers.
%F A168408 a(n) = Sum_{k=0..n} S2(n,k)*3^(n*k), where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.
%F A168408 G.f.: A(x) = Sum_{n>=0} 3^(n^2)*x^n/[Product_{k=1..n} (1-k*3^n*x)].
%F A168408 a(n) ~ 3^(n^2). - _Vaclav Kotesovec_, Jul 02 2025
%e A168408 E.g.f.: A(x) = 1 + 3*x + 90*x^2/2! + 21897*x^3/3! + 46281375*x^4/4! +...
%e A168408 A(x) = 1 + (exp(3*x) - 1) + (exp(9*x) - 1)^2/2! + (exp(27*x) - 1)^3/3! +...+ (exp(3^n*x) - 1)^n/n! +...
%e A168408 a(n) = coefficient of x^n/n! in Bell(x)^(3^n) where Bell(x) = exp(exp(x)-1):
%e A168408 Bell(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 15*x^4/4! + 52*x^5/5! + 203*x^6/6! +...+ A000110(n)*x^n/n! +...
%t A168408 Table[BellB[n, 3^n], {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 02 2025 *)
%o A168408 (PARI) {a(n)=local(infnty=n^5+10);round(exp(-3^n)*sum(k=0,infnty,(3^k*k)^n/k!))}
%o A168408 (PARI) {a(n)=n!*polcoeff(sum(k=0,n,(exp(3^k*x +x*O(x^n))-1)^k/k!),n)}
%o A168408 (PARI) {a(n)=n!*polcoeff(exp(3^n*(exp(x +x*O(x^n))-1)),n)}
%o A168408 (PARI) {S2(n,k)=(1/k!)*sum(i=0,k,(-1)^(k-i)*binomial(k,i)*i^n)}
%o A168408 {a(n)=sum(k=0,n,S2(n,k)*3^(n*k))} \\ _Paul D. Hanna_, Feb 15 2010
%o A168408 (PARI) {a(n)=polcoeff(sum(k=0,n,(3^k*x)^k/prod(j=1,k,1-j*3^k*x+x*O(x^n))),n)}
%Y A168408 Cf. A000110, A168407.
%Y A168408 Cf. A008277.
%K A168408 nonn
%O A168408 0,2
%A A168408 _Paul D. Hanna_, Nov 25 2009, Feb 16 2010