%I #24 Feb 26 2020 06:44:27

%S 1,2,10,60,492,4920,59160,828240,13253520,238563360,4771297440,

%T 104968543680,2519245713600,65500388553600,1834010896798080,

%U 55020326903942400,1760650461445075200,59862115689132556800,2155036164826415270400,81891374263403780275200

%N a(n) = (-i)^n * Integral_{x>=0} H_n(i*x) * exp(-x), where H_n(x) is n-th Hermite polynomial, i=sqrt(-1).

%H G. C. Greubel, <a href="/A277472/b277472.txt">Table of n, a(n) for n = 0..400</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>, <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>

%F a(n) = exp(1/4)*(-2*i)^n * n!*( cos(Pi*n/2)*Gamma(n/2 +1, 1/4)/Gamma(n/2 +1) + i*Gamma((n+1)/2, 1/4)*sin(Pi*n/2)/Gamma((n+1)/2) ).

%F From _Peter Luschny_, Oct 19 2016: (Start)

%F a(n) = 2^n*(n!/floor(n/2)!)*Gamma(ceiling((n+1)/2),1/4)*exp(1/4).

%F The swinging factorial A056040(n) divides a(n). (End)

%t FunctionExpand@Table[Exp[1/4] (-2 I)^n n! (Cos[Pi n/2] Gamma[n/2 + 1, 1/4]/Gamma[n/2 + 1] + I Gamma[(n + 1)/2, 1/4] Sin[Pi n/2]/Gamma[(n + 1)/2]), {n, 0, 20}]

%t FunctionExpand@Table[2^n (n!/Floor[n/2]!) Gamma[Ceiling[(n+1)/2], 1/4] Exp[1/4], {n, 0, 19}] (* _Peter Luschny_, Oct 19 2016 *)

%o (Sage)

%o def A():

%o yield 1

%o yield 2

%o a, h, f, g, n, b = 10, 5, 1, 2, 2, False

%o while True:

%o yield a

%o if b:

%o f = h

%o h = 4 * n * h + 1

%o n += 1

%o a = (a * h) // f

%o else:

%o g += 4

%o a *= g

%o b = not b

%o a = A(); print([next(a) for _ in range(20)]) # _Peter Luschny_, Oct 19 2016

%o (PARI) for(n=0, 30, print1(round(2^n*(n!/floor(n/2)!)* incgam(ceil( (n+1)/2), 1/4)*exp(1/4)), ", ")) \\ _G. C. Greubel_, Jul 12 2018

%Y Cf. A056545, A277393, A277374.

%K nonn

%O 0,2

%A _Vladimir Reshetnikov_, Oct 16 2016