OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Hermite Polynomial, Incomplete Gamma Function
Wikipedia, Hermite polynomials
FORMULA
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) ).
From Peter Luschny, Oct 19 2016: (Start)
a(n) = 2^n*(n!/floor(n/2)!)*Gamma(ceiling((n+1)/2),1/4)*exp(1/4).
The swinging factorial A056040(n) divides a(n). (End)
MATHEMATICA
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}]
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 *)
PROG
(Sage)
def A():
yield 1
yield 2
a, h, f, g, n, b = 10, 5, 1, 2, 2, False
while True:
yield a
if b:
f = h
h = 4 * n * h + 1
n += 1
a = (a * h) // f
else:
g += 4
a *= g
b = not b
a = A(); print([next(a) for _ in range(20)]) # Peter Luschny, Oct 19 2016
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Reshetnikov, Oct 16 2016
STATUS
approved