A359460 - OEIS

%I #10 Jan 03 2023 09:21:15

%S 1,1,4,18,124,1015,10446,124894,1734160,27065133,473544010,9079863496,

%T 190885380192,4332022328803,106201585772114,2781910780856250,

%U 77941165007299936,2315379935517658841,73009619250079314690,2426165226652313377828,85041434421474110745040

%N 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.

%C First negative term is a(77).

%H Paul D. Hanna, <a href="/A359460/b359460.txt">Table of n, a(n) for n = 0..300</a>

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

%F (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.

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

%e 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! + ...

%o (PARI) /* must set precision suitable for desired number of terms */

%o \p200

%o {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))}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A359461.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jan 02 2023