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A326090 E.g.f.: Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!. 4
1, 2, 6, 35, 308, 3637, 55150, 1033027, 23260536, 617066297, 18968614874, 666664879663, 26496140541700, 1179815542970053, 58388906382906390, 3189604848766578563, 191168734534622234480, 12504288586619417431921, 888401197086798248554546, 68270033412187747029025111, 5652853046029263008213465916, 502601914954325406783531231677, 47834047958592244085651443711406 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More generally, the following sums are equal:

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

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

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

LINKS

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

FORMULA

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

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

EXAMPLE

E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 35*x^3/3! + 308*x^4/4! + 3637*x^5/5! + 55150*x^6/6! + 1033027*x^7/7! + 23260536*x^8/8! + 617066297*x^9/9! + 18968614874*x^10/10! + ...

such that

A(x) = 1 + (1 + exp(x))*x + (1 + exp(2*x))^2*x^2/2! + (1 + exp(3*x))^3*x^3/3! + (1 + exp(4*x))^4*x^4/4! + (1 + exp(5*x))^5*x^5/5! + (1 + exp(6*x))^6*x^6/6! + ...

also

A(x) = exp(x) + exp(x + exp(x)*x)*x + exp(4*x + exp(2*x)*x)*x^2/2! + exp(9*x + exp(3*x)*x)*x^3/3! + exp(16*x + exp(4*x)*x)*x^4/4! + exp(25*x + exp(5*x)*x)*x^5/5! + exp(36*x + exp(6*x)*x)*x^6/6! + ...

RELATED SERIES.

Below we illustrate the following identity at specific values of x:

Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!  =  Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x ) * x^n/n!.

(1) At x = -1, the following sums are equal

S1 = Sum_{n>=0} (1 + exp(-n))^n * (-1)^n/n!,

S1 = Sum_{n>=0} exp(-n^2) * exp( -exp(-n) ) * (-1)^n/n!,

where S1 = 0.12121214669421724219987424741512642137552627624687959194...

(2) At x = -log(2), the following sums are equal

S2 = Sum_{n>=0} (1 + 1/2^n)^n * log(1/2)^n/n!,

S2 = Sum_{n>=0} 2^(-n^2) * 2^(-1/2^n) * log(1/2)^n/n!,

where S2 = 0.26746154600304489791062659014323146833150028333177021587...

PROG

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

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

for(n=0, 25, print1(a(n), ", "))

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

{a(n) = my(A = sum(m=0, n, exp(m^2*x + exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A108459, A326091, A326261, A326009.

Sequence in context: A081003 A038181 A305275 * A197973 A107868 A173399

Adjacent sequences:  A326087 A326088 A326089 * A326091 A326092 A326093

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 28 2019

STATUS

approved

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Last modified November 13 04:20 EST 2019. Contains 329085 sequences. (Running on oeis4.)