A001816 Coefficients of x^n in Hermite polynomial H_{n+4}.

12, 120, 720, 3360, 13440, 48384, 161280, 506880, 1520640, 4392960, 12300288, 33546240, 89456640, 233963520, 601620480, 1524105216, 3810263040, 9413591040, 23011000320, 55710842880, 133706022912, 318347673600, 752458137600, 1766640844800, 4122161971200

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

References

• 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.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..500
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Index entries for sequences related to Hermite polynomials.
• Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Crossrefs

Cf. A003472, A060821.

Programs

Mathematica

Table[Coefficient[HermiteH[n + 4, x], x, n], {n, 0, 25}] (* T. D. Noe, Aug 10 2012 *)
LinearRecurrence[{10,-40,80,-80,32},{12,120,720,3360,13440},30] (* Harvey P. Dale, Jul 27 2025 *)

PARI

a(n) = polcoeff(polhermite(n+4), n); \\ Michel Marcus, May 06 2022

Formula

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

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

From Amiram Eldar, May 06 2022: (Start)

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

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

Extensions

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