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%I #49 Apr 06 2023 09:22:15
%S 1,6,48,480,5760,80640,1290240,23224320,464486400,10218700800,
%T 245248819200,6376469299200,178541140377600,5356234211328000,
%U 171399494762496000,5827582821924864000,209792981589295104000,7972133300393213952000,318885332015728558080000
%N a(n) = (2*n+4)!!/4!!, related to A000165 (even double factorials).
%C Row m=4 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
%H G. C. Greubel, <a href="/A051578/b051578.txt">Table of n, a(n) for n = 0..400</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=521">Encyclopedia of Combinatorial Structures 521</a>.
%H A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
%F a(n) = (2*n+4)!!/4!!.
%F E.g.f.: 1/(1-2*x)^3.
%F a(n) ~ 2^(-1/2)*Pi^(1/2)*n^(5/2)*2^n*e^-n*n^n*{1 + 37/12*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
%F a(n) = (n+2)!*2^(n-1). - _Zerinvary Lajos_, Sep 23 2006. [corrected by _Gary Detlefs_, Apr 29 2019]
%F a(n) = 2^n*A001710(n+2). - _R. J. Mathar_, Feb 22 2008
%F From _Peter Bala_, May 26 2017: (Start)
%F a(n+1) = (2*n + 6)*a(n) with a(0) = 1.
%F O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 6*x)*A(x) - 1 with A(0) = 1.
%F G.f. as an S-fraction: A(x) = 1/(1 - 6*x/(1 - 2*x/(1 - 8*x/(1 - 4*x/(1 - 10*x/(1 - 6*x/(1 - ... - (2*n + 4)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
%F Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 6*x/(1 - 8*x/(1 - 2*x/(1 - 10*x/(1 - 4*x/(1 - 12*x/(1 - 6*x/(1 - ... - (2*n + 6)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
%F From _Amiram Eldar_, Dec 11 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 8*sqrt(e) - 12.
%F Sum_{n>=0} (-1)^n/a(n) = 8/sqrt(e) - 4. (End)
%p a:= proc(n) option remember; `if`(n=0, 1, 2*(n+2)*a(n-1)) end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Apr 29 2019
%p seq(2^(n-1)*(n+2)!, n=0..20); # _G. C. Greubel_, Nov 11 2019
%t Table[2^(n-1)(n+2)!, {n,0,20}] (* _Jean-François Alcover_, Oct 05 2019 *)
%t Table[(2n+4)!!/8,{n,0,20}] (* _Harvey P. Dale_, Apr 06 2023 *)
%o (PARI) vector(21, n, 2^(n-2)*(n+1)! ) \\ _G. C. Greubel_, Nov 11 2019
%o (Magma) [2^(n-1)*Factorial(n+2): n in [0..20]]; // _G. C. Greubel_, Nov 11 2019
%o (Sage) [2^(n-1)*factorial(n+2) for n in (0..20)] # _G. C. Greubel_, Nov 11 2019
%o (GAP) List([0..20], n-> 2^(n-1)*Factorial(n+2) ); # _G. C. Greubel_, Nov 11 2019
%Y Cf. A000165, A001147(n+1), A002866(n+1), A051577 (rows m=0..3), A051579, A051580, A051581, A051582, A051583.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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