OFFSET
0,4
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..400
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 25.
FORMULA
a(n) = (-2)^n*Sum_{k=0..n} |Stirling1(n,k)|*(-1/2)^k.
a(n) = (-2)^(n-1)*RisingFactorial(1/2, n-1).
a(n) = ((-2)^(n-1)*Gamma(n - 1/2))/sqrt(Pi).
a(n) = n!*[x^n] (1+2*x)^(1/2).
D-finite with recurrence a(n) = (3 - 2*n)*a(n-1).
a(n) = (-1)^(n-1)*(2*n-3)!! = (-1)^(n-1)*A001147(n-1).
a(2*n) = -2^(2*n-1)*RisingFactorial(1/2, 2*n-1) = -A103639(n-1).
a(2*n+1) = 4^n*RisingFactorial(1/2, 2*n) = A101485(n).
a(n) ~ -((-2*n)^n/exp(n))/(sqrt(2)*n).
Sum_{n>=0} 1/a(n) = 2 - sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erfi is the imaginary error function. - Amiram Eldar, Jan 08 2023
MAPLE
a := n -> ((-2)^(n-1)*GAMMA(n-1/2))/sqrt(Pi): seq(a(n), n=1..9);
# Alternative:
arec := proc(n) option remember: if n = 0 then 1 else
(3 - 2*n)*arec(n-1) fi end: seq(arec(n), n=0..20);
# Or:
gf := (1+2*x)^(1/2); ser := series(gf, x, 24);
seq(n!*coeff(ser, x, n), n=0..20);
MATHEMATICA
a[n_]:= (-2)^n*Sum[Abs[StirlingS1[n, k]]*(-1/2)^k, {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 19 2021 *)
Table[(-2)^(n-1)*Pochhammer[1/2, n-1], {n, 0, 30}] (* G. C. Greubel, Sep 14 2023 *)
PROG
(SageMath)
def A330797(n): return (-2)^(n-1)*rising_factorial(1/2, n-1)
[A330797(n) for n in (0..20)]
(Magma)
m:=30;
R<x>:=PowerSeriesRing(Rationals(), m+2);
A330797:= func< n | Coefficient(R!(Laplace( Sqrt(1+2*x) )), n) >;
[A330797(n): n in [0..m]]; // G. C. Greubel, Sep 14 2023
CROSSREFS
KEYWORD
sign
AUTHOR
Peter Luschny, Jan 06 2020
STATUS
approved