A326434 E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.

1, 4, 47, 895, 24450, 887803, 40818505, 2297393888, 154381810471, 12149510583583, 1102672816721422, 113974516318639363, 13277046519634998953, 1727765194711759098324, 249264545884060054668295, 39606622952407779396832791, 6891271396238954765341535650, 1306288225868329080524305347859, 268542657134280438710389415260401, 59628381166607045580114829853101712

Offset

0,2

Comments

More generally, the following sums are equal:

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

(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,

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

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.

E.g.f.: exp(-3) * Sum_{n>=0} exp(n^2*x) * exp( 2*exp(n*x) ) / n!.

FORMULAS FOR TERMS.

a(3*n+1) = 0 (mod 2), a(3*n) = 1 (mod 2), and a(3*n+2) = 1 (mod 2) for n >= 0.

Example

E.g.f.: A(x) = 1 + 4*x + 47*x^2/2! + 895*x^3/3! + 24450*x^4/4! + 887803*x^5/5! + 40818505*x^6/6! + 2297393888*x^7/7! + 154381810471*x^8/8! + 12149510583583*x^9/9! + 1102672816721422*x^10/10! + ...
such that
A(x) = exp(-3) * (1 + (exp(x) + 2) + (exp(2*x) + 2)^2/2! + (exp(3*x) + 2)^3/3! + (exp(4*x) + 2)^4/4! + (exp(5*x) + 2)^5/5! + (exp(6*x) + 2)^6/6! + ...)
also
A(x) = exp(-3) * (exp(2) + exp(x)*exp(2*exp(x)) + exp(4*x)*exp(2*exp(2*x))/2! + exp(9*x)*exp(2*exp(3*x))/3! + exp(16*x)*exp(2*exp(4*x))/4! + exp(25*x)*exp(2*exp(5*x))/5! + exp(36*x)*exp(2*exp(6*x))/6! + ...).

Prog

(PARI) /* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-3) * sum(n=0, 500, (exp(n*x +O(x^31)) + 2)^n/n! ) )))

Crossrefs

Cf. A326600, A020557, A326433, A326435, A326436, A326437.

Sequence in context: A278267 A309329 A354405 * A319833 A006438 A251665

Adjacent sequences: A326431 A326432 A326433 * A326435 A326436 A326437

Keyword

nonn

Author

Paul D. Hanna, Jul 11 2019

Status

approved