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

0,2

Links

Table of n, a(n) for n=0..11.

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)].

Example

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! +...

Prog

(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)}

Crossrefs

Cf. A000110, A168408, A277036, A008277.

Sequence in context: A171799 A251183 A158268 * A333464 A356691 A163594

Adjacent sequences: A168404 A168405 A168406 * A168408 A168409 A168410

Keyword

nonn

Author

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

Status

approved