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

%I A330797 #25 Aug 10 2025 08:14:09
%S A330797 1,1,-1,3,-15,105,-945,10395,-135135,2027025,-34459425,654729075,
%T A330797 -13749310575,316234143225,-7905853580625,213458046676875,
%U A330797 -6190283353629375,191898783962510625,-6332659870762850625,221643095476699771875,-8200794532637891559375,319830986772877770815625
%N A330797 Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.
%H A330797 G. C. Greubel, <a href="/A330797/b330797.txt">Table of n, a(n) for n = 0..400</a>
%H A330797 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 A330797 a(n) = (-2)^n*Sum_{k=0..n} |Stirling1(n,k)|*(-1/2)^k.
%F A330797 a(n) = (-2)^(n-1)*RisingFactorial(1/2, n-1).
%F A330797 a(n) = ((-2)^(n-1)*Gamma(n - 1/2))/sqrt(Pi).
%F A330797 a(n) = n!*[x^n] (1+2*x)^(1/2).
%F A330797 D-finite with recurrence a(n) = (3 - 2*n)*a(n-1).
%F A330797 a(n) = (-1)^(n-1)*(2*n-3)!! = (-1)^(n-1)*A001147(n-1).
%F A330797 a(2*n) = -2^(2*n-1)*RisingFactorial(1/2, 2*n-1) = -A103639(n-1).
%F A330797 a(2*n+1) = 4^n*RisingFactorial(1/2, 2*n) = A101485(n).
%F A330797 a(n) ~ -((-2*n)^n/exp(n))/(sqrt(2)*n).
%F A330797 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
%F A330797 O.g.f.: 1+x*2F0(1/2,1;;-2*x). - _R. J. Mathar_, Aug 10 2025
%p A330797 a := n -> ((-2)^(n-1)*GAMMA(n-1/2))/sqrt(Pi): seq(a(n), n=1..9);
%p A330797 # Alternative:
%p A330797 arec := proc(n) option remember: if n = 0 then 1 else
%p A330797 (3 - 2*n)*arec(n-1) fi end: seq(arec(n), n=0..20);
%p A330797 # Or:
%p A330797 gf := (1+2*x)^(1/2); ser := series(gf, x, 24);
%p A330797 seq(n!*coeff(ser, x, n), n=0..20);
%t A330797 a[n_]:= (-2)^n*Sum[Abs[StirlingS1[n, k]]*(-1/2)^k, {k, 0, n}];
%t A330797 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Nov 19 2021 *)
%t A330797 Table[(-2)^(n-1)*Pochhammer[1/2, n-1], {n,0,30}] (* _G. C. Greubel_, Sep 14 2023 *)
%o A330797 (SageMath)
%o A330797 def A330797(n): return (-2)^(n-1)*rising_factorial(1/2, n-1)
%o A330797 [A330797(n) for n in (0..20)]
%o A330797 (Magma)
%o A330797 m:=30;
%o A330797 R<x>:=PowerSeriesRing(Rationals(), m+2);
%o A330797 A330797:= func< n | Coefficient(R!(Laplace( Sqrt(1+2*x) )), n) >;
%o A330797 [A330797(n): n in [0..m]]; // _G. C. Greubel_, Sep 14 2023
%Y A330797 The equivalent for Stirling2 is A009235.
%Y A330797 Cf. A001147, A101485, A103639, A130534.
%Y A330797 Cf. A330799, A330800, A330802, A330803.
%K A330797 sign
%O A330797 0,4
%A A330797 _Peter Luschny_, Jan 06 2020