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%I #33 Dec 11 2022 06:01:54
%S 1,10,120,1680,26880,483840,9676800,212889600,5109350400,132843110400,
%T 3719607091200,111588212736000,3570822807552000,121407975456768000,
%U 4370687116443648000,166086110424858624000,6643444416994344960000
%N a(n) = (2*n+8)!!/8!!, related to A000165 (even double factorials).
%C Row m=8 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
%H G. C. Greubel, <a href="/A051582/b051582.txt">Table of n, a(n) for n = 0..399</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+8)!!/8!!.
%F E.g.f.: 1/(1-2*x)^5.
%F a(n) = (n+4)!*2^(n-1)/12. - _Zerinvary Lajos_, Sep 23 2006
%F From _Peter Bala_, May 26 2017: (Start)
%F a(n+1) = (2*n + 10)*a(n) with a(0) = 1.
%F O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 10*x)*A(x) - 1 with A(0) = 1.
%F G.f. as an S-fraction: A(x) = 1/(1 - 10*x/(1 - 2*x/(1 - 12*x/(1 - 4*x/(1 - 14*x/(1 - 6*x/(1 - ... - (2*n + 8)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
%F Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 10*x/(1 - 12*x/(1 - 2*x/(1 - 14*x/(1 - 4*x/(1 - 16*x/(1 - 6*x/(1 - ... - (2*n + 10)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
%F From _Amiram Eldar_, Dec 11 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 384*sqrt(e) - 632.
%F Sum_{n>=0} (-1)^n/a(n) = 384/sqrt(e) - 232. (End)
%p seq(2^n*pochhammer(5, n), n=0..20); # _G. C. Greubel_, Nov 12 2019
%t (2Range[0,20]+8)!!/8!! (* _Harvey P. Dale_, Feb 03 2013 *)
%t Table[2^n*Pochhammer[5, n], {n,0,20}] (* _G. C. Greubel_, Nov 12 2019 *)
%o (PARI) vector(20, n, n--; (n+4)!*2^(n-1)/12) \\ _Michel Marcus_, Feb 09 2015
%o (Magma) F:=Factorial; [2^n*F(n+4)/F(4): n in [0..20]]; // _G. C. Greubel_, Nov 12 2019
%o (Sage) f=factorial; [2^n*f(n+4)/f(4) for n in (0..20)] # _G. C. Greubel_, Nov 12 2019
%o (GAP) F:=Factorial;; List([0..20], n-> 2^n*F(n+4)/F(4) ); # _G. C. Greubel_, Nov 12 2019
%Y Cf. A000165, A001147(n+1), A002866(n+1).
%Y Cf. A051577, A051578, A051579, A051580, A051581 (rows m=0..7), A051583.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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