A051581 - OEIS

%I #29 Dec 11 2022 06:01:14

%S 1,9,99,1287,19305,328185,6235515,130945815,3011753745,75293843625,

%T 2032933777875,58955079558375,1827607466309625,60311046388217625,

%U 2110886623587616875,78102805072741824375,3046009397836931150625

%N a(n) = (2*n+7)!!/7!!, related to A001147 (odd double factorials).

%C Row m=7 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

%H G. C. Greubel, <a href="/A051581/b051581.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+7)!!/7!!.

%F E.g.f.: 1/(1-2*x)^(9/2).

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+9)/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 + 9)*a(n) with a(0) = 1.

%F O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 9*x)*A(x) - 1 with A(0) = 1.

%F G.f. as an S-fraction: A(x) = 1/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).

%F Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 9*x/(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 - ...)))))))))). (End)

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

%F Sum_{n>=0} 1/a(n) = 105 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 147, where erf is the error function.

%F Sum_{n>=0} (-1)^n/a(n) = 77 - 105 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2))), where erfi is the imaginary error function. (End)

%p df:=doublefactorial; seq(df(2*n+7)/df(7), n = 0..20); # _G. C. Greubel_, Nov 12 2019

%t Table[2^n*Pochhammer[9/2, n], {n,0,20}] (* _G. C. Greubel_, Nov 12 2019 *)

%o (PARI) vector(20, n, prod(j=1,n-1, 2*j+7) ) \\ _G. C. Greubel_, Nov 12 2019

%o (Magma) [1] cat [(&*[2*j+9: j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Nov 12 2019

%o (Sage) [product( (2*j+9) 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+9) ); # _G. C. Greubel_, Nov 12 2019

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

%Y Cf. A051577, A051578, A051579, A051580 (rows m=0..6), A051582, A051583.

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_