|
| |
|
|
A277393
|
|
a(n) = Integral_{x=0..infinity} H_n(x) * exp(-x), where H_n(x) is n-th Hermite polynomial.
|
|
3
|
|
|
|
1, 2, 6, 36, 300, 3000, 35880, 502320, 8038800, 144698400, 2893937760, 63666630720, 1527999802560, 39727994866560, 1112383838966400, 33371515168992000, 1067888485926662400, 36308208521506521600, 1307095506756591552000, 49669629256750478976000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
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
|
|
|
|
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) = 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)
|
|
|
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):
|
|
|
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 *)
|
|
|
PROG
|
(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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
