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Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.
4

%I #23 May 26 2024 18:39:45

%S 1,1,-1,3,-15,105,-945,10395,-135135,2027025,-34459425,654729075,

%T -13749310575,316234143225,-7905853580625,213458046676875,

%U -6190283353629375,191898783962510625,-6332659870762850625,221643095476699771875,-8200794532637891559375,319830986772877770815625

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

%H G. C. Greubel, <a href="/A330797/b330797.txt">Table of n, a(n) for n = 0..400</a>

%H Y. Alp and E. G. Kocer, <a href="https://doi.org/10.1007/s00025-024-02193-5">Exponential Almost-Riordan Arrays</a>, Results Math 79, 173 (2024). See page 25.

%F a(n) = (-2)^n*Sum_{k=0..n} |Stirling1(n,k)|*(-1/2)^k.

%F a(n) = (-2)^(n-1)*RisingFactorial(1/2, n-1).

%F a(n) = ((-2)^(n-1)*Gamma(n - 1/2))/sqrt(Pi).

%F a(n) = n!*[x^n] (1+2*x)^(1/2).

%F D-finite with recurrence a(n) = (3 - 2*n)*a(n-1).

%F a(n) = (-1)^(n-1)*(2*n-3)!! = (-1)^(n-1)*A001147(n-1).

%F a(2*n) = -2^(2*n-1)*RisingFactorial(1/2, 2*n-1) = -A103639(n-1).

%F a(2*n+1) = 4^n*RisingFactorial(1/2, 2*n) = A101485(n).

%F a(n) ~ -((-2*n)^n/exp(n))/(sqrt(2)*n).

%F 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

%p a := n -> ((-2)^(n-1)*GAMMA(n-1/2))/sqrt(Pi): seq(a(n), n=1..9);

%p # Alternative:

%p arec := proc(n) option remember: if n = 0 then 1 else

%p (3 - 2*n)*arec(n-1) fi end: seq(arec(n), n=0..20);

%p # Or:

%p gf := (1+2*x)^(1/2); ser := series(gf, x, 24);

%p seq(n!*coeff(ser, x, n), n=0..20);

%t a[n_]:= (-2)^n*Sum[Abs[StirlingS1[n, k]]*(-1/2)^k, {k, 0, n}];

%t Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Nov 19 2021 *)

%t Table[(-2)^(n-1)*Pochhammer[1/2, n-1], {n,0,30}] (* _G. C. Greubel_, Sep 14 2023 *)

%o (SageMath)

%o def A330797(n): return (-2)^(n-1)*rising_factorial(1/2, n-1)

%o [A330797(n) for n in (0..20)]

%o (Magma)

%o m:=30;

%o R<x>:=PowerSeriesRing(Rationals(), m+2);

%o A330797:= func< n | Coefficient(R!(Laplace( Sqrt(1+2*x) )), n) >;

%o [A330797(n): n in [0..m]]; // _G. C. Greubel_, Sep 14 2023

%Y The equivalent for Stirling2 is A009235.

%Y Cf. A001147, A101485, A103639, A130534.

%Y Cf. A330799, A330800, A330802, A330803.

%K sign

%O 0,4

%A _Peter Luschny_, Jan 06 2020