A051583 - OEIS

%I #26 Dec 11 2022 06:02:08

%S 1,11,143,2145,36465,692835,14549535,334639305,8365982625,

%T 225881530875,6550564395375,203067496256625,6701227376468625,

%U 234542958176401875,8678089452526869375,338445488648547905625,13876265034590464130625

%N a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).

%C Row m=9 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

%H G. C. Greubel, <a href="/A051583/b051583.txt">Table of n, a(n) for n = 0..398</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+9)!!/9!!.

%F E.g.f.: 1/(1-2*x)^(11/2).

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

%F a(n+1) = (2*n + 11)*a(n) with a(0) = 1.

%F O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 11*x)*A(x) - 1 with A(0) = 1.

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

%F Sum_{n>=0} 1/a(n) = 945 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1332, where erf is the error function.

%F 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 seq(2^n*pochhammer(11/2,n), n = 0..20); # _G. C. Greubel_, Nov 12 2019

%t (2*Range[0,20]+9)!!/945 (* _Harvey P. Dale_, Apr 10 2019 *)

%t Table[2^n*Pochhammer[11/2, n], {n,0,20}] (* _G. C. Greubel_, Nov 12 2019 *)

%o (PARI) vector(20, n, prod(j=0,n-2, 2*j+11) ) \\ _G. C. Greubel_, Nov 12 2019

%o (Magma) [1] cat [(&*[2*j+11: j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Nov 12 2019

%o (Sage) [product( (2*j+11) for j in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Nov 12 2019

%o (GAP) List([0..20], n-> Product([0..n-1], j-> 2*j+11) ); # _G. C. Greubel_, Nov 12 2019

%Y Cf. A000165, A001147(n+1), A002866(n+1), A178647.

%Y Cf. A051577, A051578, A051579, A051580, A051581, A051582 (rows m=0..8).

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_