A168407 E.g.f.: Sum_{n>=0} (exp(2^n*x) - 1)^n/n!, an analog of the Bell numbers (A000110).

1, 2, 20, 712, 91920, 44874784, 85939843136, 660213878210688, 20540390859740217600, 2592165941692975372042752, 1324271564605167892188248409088, 2730585827960928853182474922961668096

Offset: 0

Author

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

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 2*x + 20*x^2/2! + 712*x^3/3! + 91920*x^4/4! +...
A(x) = 1 + (exp(2*x) - 1) + (exp(4*x) - 1)^2/2! + (exp(8*x) - 1)^3/3! +...+ (exp(2^n*x) - 1)^n/n! +...
a(n) = coefficient of x^n/n! in Bell(x)^(2^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, A168408, A277036, A008277.

Programs

Mathematica

Table[BellB[n, 2^n], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2025 *)

PARI

{a(n)=local(infnty=n^4+10);round(exp(-2^n)*sum(k=0,infnty,(2^k*k)^n/k!))}

PARI

{a(n)=n!*polcoeff(sum(k=0,n,(exp(2^k*x +x*O(x^n))-1)^k/k!),n)}

PARI

{a(n)=n!*polcoeff(exp(2^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)*2^(n*k))}

PARI

{a(n)=polcoeff(sum(k=0,n,(2^k*x)^k/prod(j=1,k,1-j*2^k*x+x*O(x^n))),n)}

Formula

a(n) = exp(-2^n) * Sum_{k>=0} (2^k*k)^n/k!.

a(n) = [x^n/n! ] B(x)^(2^n), where B(x) = exp(exp(x) - 1) is the e.g.f. of the Bell numbers.

a(n) = Sum_{k=0..n} S2(n,k) * 2^(n*k), where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.

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

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