A326428 E.g.f.: Sum_{n>=0} (exp(n*x) + 1)^n * exp(-x*exp(n*x)) * x^n / n!.

1, 1, 3, 10, 101, 1036, 15127, 275122, 5958633, 153663832, 4616244971, 158681964574, 6183393101437, 270603517948804, 13180715982798015, 709515528436145386, 41955440825155167953, 2710310053178300043952, 190335970181019398060755, 14468433486930736773792310, 1185866281088346364062440421, 104431082548019642342105139676

Offset

0,3

Comments

More generally, the following sums are equal:

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

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

under suitable conditions; here, p = 1, q = exp(x), r = x.

Links

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

Formula

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

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

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 101*x^4/4! + 1036*x^5/5! + 15127*x^6/6! + 275122*x^7/7! + 5958633*x^8/8! + 153663832*x^9/9! + 4616244971*x^10/10! + 158681964574*x^11 + 6183393101437*x^12/12! + ...
such that
A(x) = exp(-x) + (exp(x)+1)*exp(-x*exp(x))*x + (exp(2*x)+1)^2*exp(-x*exp(2*x))*x^2/2! + (exp(3*x)+1)^3*exp(-x*exp(3*x))*x^3/3! + (exp(4*x)+1)^4*exp(-x*exp(4*x))*x^4/4! + (exp(5*x)+1)^5*exp(-x*exp(5*x))*x^5/5! + ...
also
A(x) = exp(x) + (exp(x)-1)*exp(x*exp(x))*x + (exp(2*x)-1)^2*exp(x*exp(2*x))*x^2/2! + (exp(3*x)-1)^3*exp(x*exp(3*x))*x^3/3! + (exp(4*x)-1)^4*exp(x*exp(4*x))*x^4/4! + (exp(5*x)-1)^5*exp(x*exp(5*x))*x^5/5! + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} (exp(n*x) + 1)^n * exp(-x*exp(n*x)) * x^n / n!  =  Sum_{n>=0} (exp(n*x) - 1)^n * exp(x*exp(n*x)) * x^n / n!.
(1) At x = -log(2), the following sums are equal
S1 = Sum_{n>=0} (1 + 1/2^n)^n * 2^(1/2^n) * log(1/2)^n / n!,
S1 = Sum_{n>=0} (1 - 1/2^n)^n * 2^(-1/2^n) * log(2)^n / n!,
where S1 = 0.90117245572980892640814373594212370881803992952734057573904...
(2) At x = -1, the following sums are equal
S2 = Sum_{n>=0} (1 + 1/e^n)^n * e^(1/e^n) * (-1)^n / n!,
S2 = Sum_{n>=0} (1 - 1/e^n)^n * e^(-1/e^n) / n!,
where S2 = 1.31557256964451647481459529239396855586302938480821575393879...
Compare the sums for S2 to the dual identity (cf. A326426)
S3 = Sum_{n>=0} (1 - 1/e^n)^n * e^(1/e^n) * (-1)^n / n!,
S3 = Sum_{n>=0} (1 + 1/e^n)^n * e^(-1/e^n) / n!,
where S3 = 2.11528145895544012194573695586742527663867416652856943132064...

Prog

(PARI) {a(n) = my(A = sum(m=0, n,  (exp(m*x +x*O(x^n)) + 1)^m * exp(-x*exp(m*x +x*O(x^n)) ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n,  (exp(m*x +x*O(x^n)) - 1)^m * exp(x*exp(m*x +x*O(x^n)) ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A326427, A326425, A326426.

Sequence in context: A013267 A013268 A015487 * A260813 A062013 A023372

Adjacent sequences: A326425 A326426 A326427 * A326429 A326430 A326431

Keyword

nonn

Author

Paul D. Hanna, Jul 04 2019

Status

approved