OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
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/n! * ( (1 + sqrt(n)*x)^sqrt(n) + 1/(1 - sqrt(n)*x)^sqrt(n) )/2.
(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.
(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.
(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.
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
EXAMPLE
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! + ...
MATHEMATICA
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 *)
PROG
(PARI) /* must set suitable precision for desired number of terms */
\p50
{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) )}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 03 2023
STATUS
approved