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%I M4875 N2088 #38 Sep 08 2022 08:44:29
%S 1,12,180,3360,75600,1995840,60540480,2075673600,79394515200,
%T 3352212864000,154872234316800,7771770303897600,420970891461120000,
%U 24481076457277440000,1521324036987955200000,100610229646136770560000
%N Coefficient of H_2 when expressing x^{2n} in terms of Hermite polynomials H_m.
%C a(n) = A126804(n)/2. - _Zerinvary Lajos_, Sep 21 2007
%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 Vincenzo Librandi, <a href="/A001814/b001814.txt">Table of n, a(n) for n = 1..200</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 H. E. Salzer, <a href="http://dx.doi.org/10.1090/S0025-5718-1948-0026413-5">Coefficients for expressing the first thirty powers in terms of the Hermite polynomials</a>, Math. Comp., 3 (1948), 167-169.
%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>
%F E.g.f.: x*(1 + 2x)/(1 - 4x)^(5/2).
%F a(n) = (2*n)!/(2*(n-1)!).
%F (n!/2)*binomial(2*n,n)*n or n!/2*A005430. - _Zerinvary Lajos_, Jun 06 2006
%p with(combinat):for n from 1 to 16 do printf(`%d, `,n!/2*sum(binomial(2*n, n), k=1..n)) od: # _Zerinvary Lajos_, Mar 13 2007
%p a:=n->sum((count(Permutation(n*2+2),size=n+1)),j=0..n)/2: seq(a(n), n=0..15); # _Zerinvary Lajos_, May 03 2007
%p seq(1/2*mul((n+k), k=1..n), n=0..16); # _Zerinvary Lajos_, Sep 21 2007
%t Table[(2*n)!/(2*(n-1)!),{n,1,20}] (* _Vincenzo Librandi_, Nov 22 2011 *)
%o (MuPAD) combinat::catalan(n)*binomial(n+1,2)*n! $ n = 1..16; // _Zerinvary Lajos_, Feb 15 2007
%o (Magma) [Factorial(2*n)/(2*Factorial(n-1)): n in [1..20]]; // _Vincenzo Librandi_, Nov 22 2011
%Y a(n) = A048854(n, 1) = A067147(2n, 2).
%Y Cf. A001879.
%Y Cf. A005430.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E More terms and new description from _Christian G. Bower_, Dec 18 2001