%I #31 Dec 11 2022 06:01:33

%S 1,8,80,960,13440,215040,3870720,77414400,1703116800,40874803200,

%T 1062744883200,29756856729600,892705701888000,28566582460416000,

%U 971263803654144000,34965496931549184000,1328688883398868992000

%N a(n) = (2*n+6)!!/6!!, related to A000165 (even double factorials).

%C Row m=6 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

%H G. C. Greubel, <a href="/A051580/b051580.txt">Table of n, a(n) for n = 0..400</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+6)!!/6!!.

%F E.g.f.: 1/(1-2*x)^4.

%F a(n) = n!*2^(n-4)/3, n>=3. - _Zerinvary Lajos_, Sep 23 2006

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+8)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 02 2013

%F From _Peter Bala_, May 26 2017: (Start)

%F a(n+1) = (2*n + 8)*a(n) with a(0) = 1.

%F O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 8*x)*A(x) - 1 with A(0) = 1.

%F G.f. as an S-fraction: A(x) = 1/(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 - ...))))))))) (by Stokes 1982).

%F Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 8*x/(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 - ...)))))))))). (End)

%F From _Amiram Eldar_, Dec 11 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 48*sqrt(e) - 78.

%F Sum_{n>=0} (-1)^n/a(n) = 30 - 48/sqrt(e). (End)

%p seq( mul(2*j+6, j=1..n), n=0..20); # _G. C. Greubel_, Nov 11 2019

%t Table[2^n*Pochhammer[4, n], {n,0,20}] (* _G. C. Greubel_, Nov 11 2019 *)

%t Table[(2n+6)!!/6!!,{n,0,20}] (* _Harvey P. Dale_, Mar 03 2022 *)

%o (PARI) vector(20, n, prod(j=1,n-1, 2*j+6) ) \\ _G. C. Greubel_, Nov 11 2019

%o (Magma) [1] cat [(&*[2*j+6: j in [1..n]]): n in [1..20]]; // _G. C. Greubel_, Nov 11 2019

%o (Sage) [product( (2*j+6) for j in (1..n)) for n in (0..20)] # _G. C. Greubel_, Nov 11 2019

%o (GAP) List([0..20], n-> Product([1..n], j-> 2*j+6) ); # _G. C. Greubel_, Nov 11 2019

%Y Cf. A000165, A001147(n+1), A002866(n+1), A051577, A051578, A051579 (rows m=0..5), A051581, A051582, A051583.

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_