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a(n) = coefficient of x^n/n! in A(x) = Sum_{n>=0} x^n/n! * ( (1 + sqrt(n)*x)^sqrt(n) + 1/(1 - sqrt(n)*x)^sqrt(n) )/2.
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%I #18 Jan 07 2023 10:53:02

%S 1,1,3,10,49,331,3091,36142,507585,8264917,153670771,3217628206,

%T 75150452257,1941092955127,55052488501011,1703811095028946,

%U 57225901450900801,2075951065582081417,80989170394085892451,3385153152861566082994,151069646253007978014801,7176064437477333753215491

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

%H Paul D. Hanna, <a href="/A359459/b359459.txt">Table of n, a(n) for n = 0..500</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/n! * ( (1 + sqrt(n)*x)^sqrt(n) + 1/(1 - sqrt(n)*x)^sqrt(n) )/2.

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

%F (3.a) a(n) = Sum_{k=0..n} n!/(n-k)! * ( binomial(sqrt(n-k),k) * (n-k)^(k/2) + (-1)^k * binomial(-sqrt(n-k),k) * (n-k)^(k/2) )/2.

%F (3.b) a(n) = Sum_{k=0..n} n!/k! * ( binomial(sqrt(k),n-k) * sqrt(k)^(n-k) + (-1)^(n-k) * binomial(-sqrt(k),n-k) * sqrt(k)^(n-k) )/2.

%F log(a(n)) ~ 3*n*log(n)/2 * (1 - (log(3*log(n)) + 3)/(3*log(n)) + (3*log(3*log(n)) + 2)/(9*log(n)^2)). - _Vaclav Kotesovec_, Jan 04 2023

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 49*x^4/4! + 331*x^5/5! + 3091*x^6/6! + 36142*x^7/7! + 507585*x^8/8! + 8264917*x^9/9! + 153670771*x^10/10! + ...

%t Join[{1}, Round[Table[n!*Sum[(Binomial[Sqrt[n - k], k]*(n - k)^(k/2)/(n - k)! + Binomial[n - k + Sqrt[k] - 1, n - k]*k^((n - k)/2)/k!)/2, {k, 0, n}], {n, 1, 20}]]] (* _Vaclav Kotesovec_, Jan 04 2023 *)

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

%o \p50

%o {a(n) = round( sum(k=0,n, n!/(n-k)! * ( binomial(sqrt(n-k),k)*(n-k)^(k/2) + (-1)^k*binomial(-sqrt(n-k),k)*(n-k)^(k/2) )/2) )}

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 03 2023