A277393 - OEIS

%I #47 Feb 16 2025 08:33:36

%S 1,2,6,36,300,3000,35880,502320,8038800,144698400,2893937760,

%T 63666630720,1527999802560,39727994866560,1112383838966400,

%U 33371515168992000,1067888485926662400,36308208521506521600,1307095506756591552000,49669629256750478976000

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

%C 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.

%C This is using the "physicist's" version of Hermite polynomials. - _Robert Israel_, Oct 14 2016

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

%H Robert Israel, <a href="/A277393/b277393.txt">Table of n, a(n) for n = 0..403</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>, <a href="https://mathworld.wolfram.com/HermiteDifferentialEquation.html">Hermite Differential Equation</a>

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

%F 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.

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

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

%F a(n)/n! ~ exp(-1/4) * 2^n. - _Vaclav Kotesovec_, Oct 14 2016

%F 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

%F From _Peter Luschny_, Oct 17 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).

%F 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)

%p 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

%p 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

%p f:= n -> int(orthopoly[H](n,t)*exp(-t),t=0..infinity):

%p map(f, [$0..30]); # _Robert Israel_, Oct 14 2016

%t 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}]

%t Table[Integrate[HermiteH[n, x]*Exp[-x], {x, 0, Infinity}], {n, 0, 10}] (* _G. C. Greubel_, Oct 13 2016 *)

%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 17 2016 *)

%o (Sage)

%o def A():

%o     yield 1

%o     yield 2

%o     n, a, h, i = 2, 6, -2, 2

%o     while True:

%o         yield a

%o         n += 1

%o         a *= n << 1

%o         if is_even(n):

%o             i += 4

%o             h *= -i

%o             a += h

%o H = A(); print([next(H) for _ in range(20)]) # _Peter Luschny_, Oct 16 2016

%Y Cf. A001907, A056040, A277472.

%K nonn

%O 0,2

%A _Vladimir Reshetnikov_, Oct 12 2016