A357790 - OEIS

%I #16 Jan 04 2023 12:39:31

%S 1,1,2,9,48,305,2280,19537,188608,2024577,23911200,308049401,

%T 4298093184,64555255921,1038311141504,17803434637185,324148992092160,

%U 6245040776838017,126919440612205056,2713418986517310313,60871624993766717440,1429679116231319002161

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

%H Paul D. Hanna, <a href="/A357790/b357790.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 as follows.

%F (1) A(x) = Sum_{n>=0} x^n * cosh(sqrt(n)*x).

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

%F (3) A(x) = Sum_{n>=0} x^n * Sum_{k>=0} n^k * x^(2*k)/(2*k)!.

%F (4) a(n) = Sum_{k=0..floor(n/2)} (n - 2*k)^k * n! / (2*k)!.

%F a(n) ~ sqrt(Pi/2) * n^(n + 1/2) / exp(n - sqrt(n) + 1/2). - _Vaclav Kotesovec_, Jan 04 2023

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 305*x^5/5! + 2280*x^6/6! + 19537*x^7/7! + 188608*x^8/8! + 2024577*x^9/9! + 23911200*x^10/10! + ...

%e RELATED SERIES.

%e log(A(x)) = x + x^2/2! + 5*x^3/3! + 18*x^4/4! + 89*x^5/5! + 480*x^6/6! + 3037*x^7/7! + 21392*x^8/8! + 170865*x^9/9! + 1527840*x^10/10! + 15377141*x^11/11! + 172943232*x^12/12! + ...

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

%o (PARI) {a(n) = sum(k=0, n\2, (n-2*k)^k * n!/(2*k)! )}

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

%o (PARI) {a(n) = my(A=1);

%o A = sum(m=0,n, x^m * sum(k=0,(n-m)\2+1, m^k * x^(2*k)/(2*k)! ) +x*O(x^n));

%o n!*polcoeff(A,n)}

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 01 2023