A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.

1, 2, 18, 314, 8434, 314362, 15278642, 928696442, 68509258098, 5995762219514, 611538502747826, 71656036268121978, 9532232740451770866, 1425414297318661354746, 237588200534263288095538, 43821269448954050939558522, 8887255081413035850889914994, 1970841722610600810208914571258, 475544555000142351430865220032434, 124299766720856839788225909600114042, 35056463298676734373530025799446104818

Offset

0,2

Comments

More generally, the following sums are equal:

Sum_{n>=0} (p + q^n)^n * r^n/n! =

Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = exp(x) with p = -1, r = 2.

In general, let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =

Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);

here, F(x) = exp(x), q = exp(x), p = -1, r = 2, m = 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.

E.g.f.: Sum_{n>=0} 2^n * exp(n^2*x) * exp( -2*exp(n*x) ) / n!.

O.g.f.: Sum_{n>=0} 2^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).

a(n) = Sum_{k=0..n} 2^k * k^n * Stirling2(n,k).

Example

E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 314*x^3/3! + 8434*x^4/4! + 314362*x^5/5! + 15278642*x^6/6! + 928696442*x^7/7! + 68509258098*x^8/8! + 5995762219514*x^9/9! + 611538502747826*x^10/10! + ...
such that
A(x) = 1 + 2*(exp(x) - 1) + 2^2*(exp(2*x) - 1)^2/2! + 2^3*(exp(3*x) - 1)^3/3! + 2^4*(exp(4*x) - 1)^4/4! + 2^5*(exp(5*x) - 1)^5/5! + 2^6*(exp(6*x) - 1)^6/6! + ...
also
A(x) = exp(-2) + 2*exp(x)*exp(-2*exp(x)) + 2^2*exp(4*x)*exp(-2*exp(2*x))/2! + 2^3*exp(9*x)*exp(-2*exp(3*x))/3! + 2^4*exp(16*x)*exp(-2*exp(4*x))/4! + 2^5*exp(25*x)*exp(-2*exp(5*x))/5! + 2^6*exp(36*x)*exp(-2*exp(6*x))/6! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 2*x + 18*x^2 + 314*x^3 + 8434*x^4 + 314362*x^5 + 15278642*x^6 + 928696442*x^7 + 68509258098*x^8 + 5995762219514*x^9 + ...
such that
B(x) = 1 + 2*x/(1-x) + 2^2*2^2*x^2/((1-2*x)*(1-4*x)) + 2^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 2^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 2^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!  =  Sum_{n>=0} 2^n * exp(n^2*x) * exp( -2*exp(n*x) ) / n!.
(1) At x = -1, the following sums are equal
S1 = Sum_{n>=0} (-2)^n * (1 - exp(-n))^n / n!,
S1 = Sum_{n>=0} 2^n * exp(-n^2) * exp( -2*exp(-n) ) / n!,
where S1 = 0.51596189603321982013621912500044621350106513780391377129738...
(2) At x = -2, the following sums are equal
S2 = Sum_{n>=0} (-2)^n * (1 - exp(-2*n))^n / n!,
S2 = Sum_{n>=0} 2^n * exp(-2*n^2) * exp( -2*exp(-2*n) ) / n!,
where S2 = 0.34246794778612083304129071190905516612972983097016819355092...
(3) At x = -log(2), the following sums are equal
S3 = Sum_{n>=0} 2^(-n*(n-1)) * (2^n - 1)^n * (-1)^n / n!,
S3 = Sum_{n>=0} 2^(-n*(n-1)) * exp( -1/2^(n-1) ) / n!,
where S3 = 0.58106816860114387883649557314841837351794236167582918403231...

Mathematica

Flatten[{1, Table[Sum[2^k * k^n * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 09 2019 *)

Prog

(PARI) {a(n) = sum(k=0, n, 2^k * k^n * stirling(n, k, 2) )}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n! */
{a(n) = n! * polcoeff(sum(m=0, n, 2^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* O.g.f.: Sum_{n>=0} 2^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
{a(n) = polcoeff(sum(m=0, n, 2^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A108459, A326271, A326288.

Sequence in context: A370056 A179497 A296837 * A227325 A350291 A087215

Adjacent sequences: A326267 A326268 A326269 * A326271 A326272 A326273

Keyword

nonn

Author

Paul D. Hanna, Jun 28 2019

Status

approved