A330797 Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.

1, 1, -1, 3, -15, 105, -945, 10395, -135135, 2027025, -34459425, 654729075, -13749310575, 316234143225, -7905853580625, 213458046676875, -6190283353629375, 191898783962510625, -6332659870762850625, 221643095476699771875, -8200794532637891559375, 319830986772877770815625

Offset: 0

Author

Peter Luschny, Jan 06 2020

Keywords

• sign

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.

Crossrefs

The equivalent for Stirling2 is A009235.

Cf. A001147, A101485, A103639, A130534.

Cf. A330799, A330800, A330802, A330803.

Programs

Magma

m:=30;
R:=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

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 *)

SageMath

def A330797(n): return (-2)^(n-1)*rising_factorial(1/2, n-1)
[A330797(n) for n in (0..20)]

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

O.g.f.: 1+x*2F0(1/2,1;;-2*x). - R. J. Mathar, Aug 10 2025