A326427 E.g.f.: Sum_{n>=0} ((1+x)^n + 1)^n * exp(-x*(1+x)^n) * x^n / n!.

1, 1, 3, 7, 85, 621, 9391, 153763, 2985417, 72857305, 2023660891, 64463660031, 2355689179933, 96585061458757, 4411550668276935, 223553495946198811, 12468589636275471121, 760686363827937889713, 50544936457434070715827, 3641703038385219667465975, 283296615496743105794529381, 23710312648334283124382200861, 2128138462625913049294100820703

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 = 1+x, r = x.

Links

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

Formula

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

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

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 7*x^3/3! + 85*x^4/4! + 621*x^5/5! + 9391*x^6/6! + 153763*x^7/7! + 2985417*x^8/8! + 72857305*x^9/9! + 2023660891*x^10/10! + ...
such that
A(x) = exp(-x) + ((1+x)+1)*exp(-x*(1+x))*x + ((1+x)^2+1)^2*exp(-x*(1+x)^2)*x^2/2! + ((1+x)^3+1)^3*exp(-x*(1+x)^3)*x^3/3! + ((1+x)^4+1)^4*exp(-x*(1+x)^4)*x^4/4! + ((1+x)^5+1)^5*exp(-x*(1+x)^5)*x^5/5! + ((1+x)^5+1)^6*exp(-x*(1+x)^6)*x^6/6! + ....
also
A(x) = exp(x) + ((1+x)-1)*exp(x*(1+x))*x + ((1+x)^2-1)^2*exp(x*(1+x)^2)*x^2/2! + ((1+x)^3-1)^3*exp(x*(1+x)^3)*x^3/3! + ((1+x)^4-1)^4*exp(x*(1+x)^4)*x^4/4! + ((1+x)^5-1)^5*exp(x*(1+x)^5)*x^5/5! + ((1+x)^5-1)^6*exp(x*(1+x)^6)*x^6/6! + ....
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} ((1+x)^n + 1)^n * exp(-x*(1+x)^n) * x^n / n!  =  Sum_{n>=0} ((1+x)^n - 1)^n * exp(+x*(1+x)^n) * x^n / n!.
(1) At x = -1/2, the following sums are equal
S1 = Sum_{n>=0} (1 + 1/2^n)^n * exp(1/2^(n+1)) * (-1/2)^n / n!,
S1 = Sum_{n>=0} (1 - 1/2^n)^n * exp(-1/2^(n+1)) * (1/2^n) / n!,
where S1 = 0.87858219381273485321461030911688165612390011991090257894390...
(2) At x = -2/3, the following sums are equal
S2 = Sum_{n>=0} (1 + 1/3^n)^n * exp(2/3^(n+1)) * (-2/3)^n / n!,
S2 = Sum_{n>=0} (1 - 1/3^n)^n * exp(-2/3^(n+1)) * (2/3)^n / n!,
where S2 = 1.08434104867591642538828152304288635285016437420102640250465...

Prog

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

Crossrefs

Cf. A326428, A326425, A326426.

Sequence in context: A001531 A219588 A219375 * A064118 A041705 A137130

Adjacent sequences: A326424 A326425 A326426 * A326428 A326429 A326430

Keyword

nonn

Author

Paul D. Hanna, Jul 04 2019

Status

approved