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Hermite numbers.
10

%I #82 Sep 08 2022 08:45:05

%S 1,0,-2,0,12,0,-120,0,1680,0,-30240,0,665280,0,-17297280,0,518918400,

%T 0,-17643225600,0,670442572800,0,-28158588057600,0,1295295050649600,0,

%U -64764752532480000,0,3497296636753920000,0,-202843204931727360000,0

%N Hermite numbers.

%C |a(n)| is the number of sets of ordered pairs of n labeled elements. - _Steven Finch_, Nov 14 2021

%C |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

%C Self-convolution gives A076729(n)*(-1)^n interleaved with zeros. - _Vladimir Reshetnikov_, Oct 11 2016

%C Named after the French mathematician Charles Hermite (1822-1901). - _Amiram Eldar_, Jun 06 2021

%H G. C. Greubel, <a href="/A067994/b067994.txt">Table of n, a(n) for n = 0..730</a>

%H Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermiteNumber.html">Hermite Number</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hermite_number">Hermite number</a>.

%F E.g.f.: exp(-x^2). - _Vladeta Jovovic_, Aug 24 2002

%F a(n) = (-1)^(n/2)*n!/(n/2)! if n is even, 0 otherwise. - _Mitch Harris_, Feb 01 2006

%F a(n) = -(2*n-2)*a(n-2). - _Alexander Karpov_, Jul 24 2017

%F 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

%F 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

%F 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

%F 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

%F a(2*k) = A097388(k), a(2*k+1) = 0. - _Joerg Arndt_, Oct 12 2016

%F From _Peter Luschny_, Nov 14 2021: (Start)

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

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

%e From _Steven Finch_, Nov 14 2021: (Start)

%e |a(4)| = 12 because the sets of ordered pairs for n = 4 are

%e {(1,2),(3,4)}, {(2,1),(3,4)}, {(1,2),(4,3)}, {(2,1),(4,3)},

%e {(1,3),(2,4)}, {(3,1),(2,4)}, {(1,3),(4,2)}, {(3,1),(4,2)},

%e {(1,4),(3,2)}, {(4,1),(3,2)}, {(1,4),(2,3)}, {(4,1),(2,3)}. (End)

%p A067994 := n -> pochhammer(-n, n/2):

%p seq(A067994(n), n = 0..31); # _Peter Luschny_, Nov 14 2021

%t HermiteH[Range[0,50], 0]

%t With[{nmax=50}, CoefficientList[Series[Exp[-x^2], {x,0,nmax}],x]*Range[0, nmax]!] (* _G. C. Greubel_, Jun 09 2018 *)

%o (PARI) a(n) = polhermite(n, 0); \\ _Michel Marcus_, Feb 27 2015

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(-x^2))) \\ _G. C. Greubel_, Jun 09 2018

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jun 09 2018

%Y Cf. A097388 (same sequence without zeros).

%Y Cf. A001813, A076729, A126869, A081123.

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

%K sign

%O 0,3

%A _Eric W. Weisstein_, Feb 07 2002