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 A067994 Hermite numbers. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 LINKS G. C. Greubel, Table of n, a(n) for n = 0..730 Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021. Eric Weisstein's World of Mathematics, Hermite Number. Wikipedia, Hermite number. 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) EXAMPLE 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) 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 *) PROG (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 (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 CROSSREFS Cf. A097388 (same sequence without zeros). Cf. A001813, A076729, A126869, A081123. Cf. A101109 (ordered triples instead of ordered pairs). Sequence in context: A292496 A285480 A156431 * A236219 A143246 A286198 Adjacent sequences:  A067991 A067992 A067993 * A067995 A067996 A067997 KEYWORD sign,changed AUTHOR Eric W. Weisstein, Feb 07 2002 STATUS approved

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Last modified December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)