A359460 a(n) = coefficient of x^n/n! in A(x) = Sum_{n>=0} x^n * ( (exp(sqrt(n)*x) + x)^sqrt(n) + exp(n*x)/(1 + x*exp(sqrt(n)*x))^sqrt(n) )/2.

1, 1, 4, 18, 124, 1015, 10446, 124894, 1734160, 27065133, 473544010, 9079863496, 190885380192, 4332022328803, 106201585772114, 2781910780856250, 77941165007299936, 2315379935517658841, 73009619250079314690, 2426165226652313377828, 85041434421474110745040

Offset

0,3

Comments

First negative term is a(77).

Links

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

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.

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

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

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 124*x^4/4! + 1015*x^5/5! + 10446*x^6/6! + 124894*x^7/7! + 1734160*x^8/8! + 27065133*x^9/9! + 473544010*x^10/10! + 9079863496*x^11/11! + 190885380192*x^12/12! + ...

Prog

(PARI) /* must set precision suitable for desired number of terms */
\p200
{a(n) = my(A=1); A = sum(m=0, n, x^m/2 * ( (exp(sqrt(m)*x +x*O(x^n)) + x)^sqrt(m) + exp(m*x +x*O(x^n))/(1 + x*exp(sqrt(m)*x +x*O(x^n)) )^sqrt(m) ) ); round(n!*polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A359461.

Sequence in context: A317377 A332748 A215691 * A073511 A108704 A001423

Adjacent sequences: A359457 A359458 A359459 * A359461 A359462 A359463

Keyword

sign

Author

Paul D. Hanna, Jan 02 2023

Status

approved