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

1, 3, 90, 21897, 46281375, 882516062106, 153201395082609531, 241604140428719375972139, 3448358784659838019970862469260, 444238039567848645977924129826080612043

Offset: 0

Author

Paul D. Hanna, Nov 25 2009, Feb 16 2010

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 3*x + 90*x^2/2! + 21897*x^3/3! + 46281375*x^4/4! +...
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! +...
a(n) = coefficient of x^n/n! in Bell(x)^(3^n) where Bell(x) = exp(exp(x)-1):
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! +...
		

Crossrefs

Cf. A000110, A168407.

Cf. A008277.

Programs

Mathematica

Table[BellB[n, 3^n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2025 *)

PARI

{a(n)=local(infnty=n^5+10);round(exp(-3^n)*sum(k=0,infnty,(3^k*k)^n/k!))}

PARI

{a(n)=n!*polcoeff(sum(k=0,n,(exp(3^k*x +x*O(x^n))-1)^k/k!),n)}

PARI

{a(n)=n!*polcoeff(exp(3^n*(exp(x +x*O(x^n))-1)),n)}

PARI

{S2(n,k)=(1/k!)*sum(i=0,k,(-1)^(k-i)*binomial(k,i)*i^n)}
{a(n)=sum(k=0,n,S2(n,k)*3^(n*k))} \\ Paul D. Hanna, Feb 15 2010

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)}

Formula

a(n) = exp(-3^n)*Sum_{k>=0} (3^k*k)^n/k!.

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.

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.

G.f.: A(x) = Sum_{n>=0} 3^(n^2)*x^n/[Product_{k=1..n} (1-k*3^n*x)].

a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jul 02 2025