This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A051583 #29 Apr 19 2025 04:29:35
%S A051583 1,11,143,2145,36465,692835,14549535,334639305,8365982625,
%T A051583 225881530875,6550564395375,203067496256625,6701227376468625,
%U A051583 234542958176401875,8678089452526869375,338445488648547905625,13876265034590464130625
%N A051583 a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).
%C A051583 Row m=9 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
%H A051583 G. C. Greubel, <a href="/A051583/b051583.txt">Table of n, a(n) for n = 0..398</a>
%H A051583 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 A051583 a(n) = (2*n+9)!!/9!!.
%F A051583 E.g.f.: 1/(1-2*x)^(11/2).
%F A051583 From _Peter Bala_, May 26 2017: (Start)
%F A051583 a(n+1) = (2*n + 11)*a(n) with a(0) = 1.
%F A051583 O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 11*x)*A(x) - 1 with A(0) = 1.
%F A051583 G.f. as an S-fraction: A(x) = 1/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
%F A051583 Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 11*x/(1 - 13*x/(1 - 2*x/(1 - 15*x/(1 - 4*x/(1 - 17*x/(1 - 6*x/(1 - ... - (2*n + 11)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
%F A051583 From _Amiram Eldar_, Dec 11 2022: (Start)
%F A051583 Sum_{n>=0} 1/a(n) = 945 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1332, where erf is the error function.
%F A051583 Sum_{n>=0} (-1)^n/a(n) = 945 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 684, where erfi is the imaginary error function. (End)
%p A051583 seq(2^n*pochhammer(11/2,n), n = 0..20); # _G. C. Greubel_, Nov 12 2019
%t A051583 (2*Range[0,20]+9)!!/945 (* _Harvey P. Dale_, Apr 10 2019 *)
%t A051583 Table[2^n*Pochhammer[11/2, n], {n,0,20}] (* _G. C. Greubel_, Nov 12 2019 *)
%o A051583 (PARI) vector(20, n, prod(j=0,n-2, 2*j+11) ) \\ _G. C. Greubel_, Nov 12 2019
%o A051583 (Magma) [1] cat [(&*[2*j+11: j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Nov 12 2019
%o A051583 (Sage) [product( (2*j+11) for j in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Nov 12 2019
%o A051583 (GAP) List([0..20], n-> Product([0..n-1], j-> 2*j+11) ); # _G. C. Greubel_, Nov 12 2019
%Y A051583 Cf. A000165, A001147(n+1), A002866(n+1), A178647.
%Y A051583 Cf. A051577, A051578, A051579, A051580, A051581, A051582 (rows m=0..8).
%K A051583 easy,nonn
%O A051583 0,2
%A A051583 _Wolfdieter Lang_