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Coefficients of x^n in Hermite polynomial H_{n+4}
(Formerly M4862 N2078)
2

%I M4862 N2078 #31 Jun 25 2023 02:41:40

%S 12,120,720,3360,13440,48384,161280,506880,1520640,4392960,12300288,

%T 33546240,89456640,233963520,601620480,1524105216,3810263040,

%U 9413591040,23011000320,55710842880,133706022912,318347673600,752458137600,1766640844800,4122161971200

%N Coefficients of x^n in Hermite polynomial H_{n+4}

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001816/b001816.txt">Table of n, a(n) for n = 0..500</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10, -40, 80, -80, 32).

%F a(n) = 12*A003472(n+4) = A060821(4+n, n).

%F G.f.: 12 ( 1 - 2 x )^(-5).

%F From _Amiram Eldar_, May 06 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 5/9 - 2*log(2)/3.

%F Sum_{n>=0} (-1)^n/a(n) = 18*log(3/2) - 65/9. (End)

%t Table[Coefficient[HermiteH[n + 4, x], x, n], {n, 0, 25}] (* _T. D. Noe_, Aug 10 2012 *)

%o (PARI) a(n) = polcoeff(polhermite(n+4), n); \\ _Michel Marcus_, May 06 2022

%Y Cf. A003472, A060821.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from Larry Reeves (larryr(AT)acm.org), Jan 29 2001