A357790 a(n) = coefficient of x^n/n! in A(x) = Sum_{n>=0} x^n * cosh(sqrt(n)*x).

1, 1, 2, 9, 48, 305, 2280, 19537, 188608, 2024577, 23911200, 308049401, 4298093184, 64555255921, 1038311141504, 17803434637185, 324148992092160, 6245040776838017, 126919440612205056, 2713418986517310313, 60871624993766717440, 1429679116231319002161

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 as follows.

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

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

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

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

a(n) ~ sqrt(Pi/2) * n^(n + 1/2) / exp(n - sqrt(n) + 1/2). - Vaclav Kotesovec, Jan 04 2023

Example

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! + ...
RELATED SERIES.
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! + ...

Mathematica

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

Prog

(PARI) {a(n) = sum(k=0, n\2, (n-2*k)^k * n!/(2*k)! )}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1);
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));
n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A214404 A074143 A198892 * A205571 A354312 A052826

Adjacent sequences: A357787 A357788 A357789 * A357791 A357792 A357793

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2023

Status

approved