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
Keywords
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
Programs
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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
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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)
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