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).

1, 2, 10, 60, 492, 4920, 59160, 828240, 13253520, 238563360, 4771297440, 104968543680, 2519245713600, 65500388553600, 1834010896798080, 55020326903942400, 1760650461445075200, 59862115689132556800, 2155036164826415270400, 81891374263403780275200

Offset: 0

Author

Vladimir Reshetnikov, Oct 16 2016

Keywords

• nonn

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

Crossrefs

Cf. A056545, A277393, A277374.

Programs

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 *)

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

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

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)