login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!. 4
1, 2, 18, 314, 8434, 314362, 15278642, 928696442, 68509258098, 5995762219514, 611538502747826, 71656036268121978, 9532232740451770866, 1425414297318661354746, 237588200534263288095538, 43821269448954050939558522, 8887255081413035850889914994, 1970841722610600810208914571258, 475544555000142351430865220032434, 124299766720856839788225909600114042, 35056463298676734373530025799446104818 (list; graph; refs; listen; history; text; internal format)
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: A192555 A179497 A296837 * A227325 A087215 A229490

Adjacent sequences:  A326267 A326268 A326269 * A326271 A326272 A326273

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 28 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 01:30 EST 2021. Contains 349617 sequences. (Running on oeis4.)