A277393 a(n) = Integral_{x=0..infinity} H_n(x) * exp(-x), where H_n(x) is n-th Hermite polynomial.

1, 2, 6, 36, 300, 3000, 35880, 502320, 8038800, 144698400, 2893937760, 63666630720, 1527999802560, 39727994866560, 1112383838966400, 33371515168992000, 1067888485926662400, 36308208521506521600, 1307095506756591552000, 49669629256750478976000

Offset: 0

Author

Vladimir Reshetnikov, Oct 12 2016

Keywords

• nonn

Comments

Hermite polynomials can be generalized to non-integer or even complex indexes using their representation as a contour integral (or as a solution to a differential equation), in which case the first formula for a(n) and the functional relation (recurrence) given below remain valid for all complex n.

This is using the "physicist's" version of Hermite polynomials. - Robert Israel, Oct 14 2016

References

• George E. Andrews, Richard Askey, Ranjan Roy, Special Functions, Cambridge University Press (p.278 for Hermite polynomials).

Links

• Robert Israel, Table of n, a(n) for n = 0..403
• Eric Weisstein's World of Mathematics, Hermite Polynomial, Hermite Differential Equation
• Wikipedia, Hermite polynomials

Crossrefs

Cf. A001907, A056040, A277472.

Programs

Maple

a := proc(n) 4^x*sqrt(Pi)*exp(-1/4)*(GAMMA(1+x/2, -1/4)/((-1)^(x/2)*GAMMA((1-x)/2)) + x*GAMMA((x+1)/2, -1/4)/(2*(-1)^((x-1)/2)*GAMMA(1-x/2))); simplify(limit (%,x=n)) end: seq(a(n),n=0..9); # Peter Luschny, Oct 14 2016
a := n -> (cos(Pi*n/2)*GAMMA((n+1)/2)*GAMMA(n/2+1, -1/4) + I*sin(Pi*n/2)*GAMMA(n/2+1)*GAMMA((n+1)/2, -1/4))/(sqrt(Pi)*exp(1/4)*(I/4)^n): seq(a(n), n=0..20); # Vladimir Reshetnikov, Oct 14 2016
f:= n -> int(orthopoly[H](n,t)*exp(-t),t=0..infinity):
map(f, [$0..30]); # Robert Israel, Oct 14 2016

Mathematica

FunctionExpand@Table[4^n Sqrt[Pi] Exp[-1/4] (Gamma[n/2 + 1, -1/4]/((-1)^(n/2) Gamma[(1 - n)/2]) + n  Gamma[(n + 1)/2, -1/4]/(2 (-1)^((n - 1)/2) Gamma[1 - n/2])), {n, 0, 20}]
Table[Integrate[HermiteH[n, x]*Exp[-x], {x, 0, Infinity}], {n, 0, 10}] (* G. C. Greubel, Oct 13 2016 *)
FunctionExpand@Table[2^n*(n!/Floor[n/2]!)*Gamma[Ceiling[(n+1)/2],-1/4]*Exp[-1/4], {n,0,19}] (* Peter Luschny, Oct 17 2016 *)

Sage

def A():
    yield 1
    yield 2
    n, a, h, i = 2, 6, -2, 2
    while True:
        yield a
        n += 1
        a *= n << 1
        if is_even(n):
            i += 4
            h *= -i
            a += h
H = A(); print([next(H) for  in range(20)]) # _Peter Luschny, Oct 16 2016

Formula

a(n) = 4^n*sqrt(Pi)*exp(-1/4)*(Gamma(1+n/2, -1/4)/((-1)^(n/2)*Gamma((1-n)/2)) + n*Gamma((n+1)/2, -1/4)/(2*(-1)^((n-1)/2)*Gamma(1-n/2))), assuming that 1/Gamma(z) is an entire function of z having zeros at nonpositive integer arguments.

Recurrence: 2*((n+1)*a(n) + 2*n*(n-1)*a(n-2)) = 2*n*a(n-1) + a(n+1).

E.g.f.: exp(-x^2)/(1-2*x).

a(n)/n! ~ exp(-1/4) * 2^n. - Vaclav Kotesovec, Oct 14 2016

a(2*n) = 2^n*(2*n-1)!!*A001907(n), a(2*n+1) = 2^(n+1)*(2*n+1)!!*A001907(n). - Vladimir Reshetnikov, Oct 14 2016

From Peter Luschny, Oct 17 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).

Recurrence: If n is odd then a(n) = a(n-1)*n*2 else a(n) = a(n-1)*n*2 + (-1)^[n/2]* n!/[n/2]!. See the Sage implementation. (End)