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A277472 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).

%I A277472 #25 Feb 16 2025 08:33:36
%S A277472 1,2,10,60,492,4920,59160,828240,13253520,238563360,4771297440,
%T A277472 104968543680,2519245713600,65500388553600,1834010896798080,
%U A277472 55020326903942400,1760650461445075200,59862115689132556800,2155036164826415270400,81891374263403780275200
%N A277472 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 A277472 G. C. Greubel, <a href="/A277472/b277472.txt">Table of n, a(n) for n = 0..400</a>
%H A277472 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>, <a href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>
%H A277472 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>
%F A277472 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 A277472 From _Peter Luschny_, Oct 19 2016: (Start)
%F A277472 a(n) = 2^n*(n!/floor(n/2)!)*Gamma(ceiling((n+1)/2),1/4)*exp(1/4).
%F A277472 The swinging factorial A056040(n) divides a(n). (End)
%t A277472 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 A277472 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 A277472 (Sage)
%o A277472 def A():
%o A277472     yield 1
%o A277472     yield 2
%o A277472     a, h, f, g, n, b = 10, 5, 1, 2, 2, False
%o A277472     while True:
%o A277472         yield a
%o A277472         if b:
%o A277472             f = h
%o A277472             h = 4 * n * h + 1
%o A277472             n += 1
%o A277472             a = (a * h) // f
%o A277472         else:
%o A277472             g += 4
%o A277472             a *= g
%o A277472         b = not b
%o A277472 a = A(); print([next(a) for _ in range(20)]) # _Peter Luschny_, Oct 19 2016
%o A277472 (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 A277472 Cf. A056545, A277393, A277374.
%K A277472 nonn
%O A277472 0,2
%A A277472 _Vladimir Reshetnikov_, Oct 16 2016