A067994 Hermite numbers.

1, 0, -2, 0, 12, 0, -120, 0, 1680, 0, -30240, 0, 665280, 0, -17297280, 0, 518918400, 0, -17643225600, 0, 670442572800, 0, -28158588057600, 0, 1295295050649600, 0, -64764752532480000, 0, 3497296636753920000, 0, -202843204931727360000, 0

Offset: 0

Author

Eric W. Weisstein, Feb 07 2002

Keywords

• sign

Comments

|a(n)| is the number of sets of ordered pairs of n labeled elements. - Steven Finch, Nov 14 2021

|a(n)| is the number of square roots of any permutation in S_{2n} whose disjoint cycle decomposition consists of n transpositions, n > 0. For n=2, permutation (1,2)(3,4) in S_4 has exactly |a(2)|=2 square roots: (1,3,2,4) and (1,4,2,3). - Luis Manuel Rivera Martínez, Feb 25 2015

Self-convolution gives A076729(n)*(-1)^n interleaved with zeros. - Vladimir Reshetnikov, Oct 11 2016

Named after the French mathematician Charles Hermite (1822-1901). - Amiram Eldar, Jun 06 2021

Examples

			From _Steven Finch_, Nov 14 2021: (Start)
|a(4)| = 12 because the sets of ordered pairs for n = 4 are
  {(1,2),(3,4)}, {(2,1),(3,4)}, {(1,2),(4,3)}, {(2,1),(4,3)},
  {(1,3),(2,4)}, {(3,1),(2,4)}, {(1,3),(4,2)}, {(3,1),(4,2)},
  {(1,4),(3,2)}, {(4,1),(3,2)}, {(1,4),(2,3)}, {(4,1),(2,3)}. (End)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..730
• Giuseppe Dattoli, Subuhi Khan, and Ujair Ahmad, Hermite numbers and new families of polynomials, arXiv:2503.14930 [math.NT], 2025.
• Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
• Eric Weisstein's World of Mathematics, Hermite Number.
• Wikipedia, Hermite number.

Crossrefs

Cf. A097388 (same sequence without zeros).

Cf. A001813, A076729, A126869, A081123.

Cf. A101109 (ordered triples instead of ordered pairs).

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

Maple

A067994 := n -> pochhammer(-n, n/2):
seq(A067994(n), n = 0..31); # Peter Luschny, Nov 14 2021

Mathematica

HermiteH[Range[0,50], 0]
With[{nmax=50}, CoefficientList[Series[Exp[-x^2], {x,0,nmax}],x]*Range[0, nmax]!] (* G. C. Greubel, Jun 09 2018 *)

PARI

a(n) = polhermite(n, 0); \\ Michel Marcus, Feb 27 2015

PARI

x='x+O('x^30); Vec(serlaplace(exp(-x^2))) \\ G. C. Greubel, Jun 09 2018

Formula

E.g.f.: exp(-x^2). - Vladeta Jovovic, Aug 24 2002

a(n) = (-1)^(n/2)*n!/(n/2)! if n is even, 0 otherwise. - Mitch Harris, Feb 01 2006

a(n) = -(2*n-2)*a(n-2). - Alexander Karpov, Jul 24 2017

E.g.f.: U(0) where U(k) = 1 - x^2/((2*k+1) - x^2*(2*k+1)/(x^2 - 2*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2012

G.f.: 1/G(0) where G(k) = 1 + 2*x^2*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2012

E.g.f.: E(0)/(1+x) where E(k) = 1 + x/(1 - x/(x - (k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013

E.g.f.: E(0)-1, where E(k) = 2 - x^2/(2*k+1 + x^2/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013

a(2*k) = A097388(k), a(2*k+1) = 0. - Joerg Arndt, Oct 12 2016

From Peter Luschny, Nov 14 2021: (Start)

a(n) = A057077(n)*A126869(n)*A081123(n). In particular, a(n) is divisible by floor(n/2)!.

a(n) = Pochhammer(-n, n/2). (End)