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A051577 a(n) = (2*n + 3)!!/3 = A001147(n+2)/3.

%I A051577 #69 Jan 18 2024 07:50:04
%S A051577 1,5,35,315,3465,45045,675675,11486475,218243025,4583103525,
%T A051577 105411381075,2635284526875,71152682225625,2063427784543125,
%U A051577 63966261320836875,2110886623587616875,73881031825566590625,2733598177545963853125,106610328924292590271875
%N A051577 a(n) = (2*n + 3)!!/3 = A001147(n+2)/3.
%C A051577 Row m = 3 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
%H A051577 G. C. Greubel, <a href="/A051577/b051577.txt">Table of n, a(n) for n = 0..250</a>
%H A051577 H. W. Gould, Harris Kwong, and Jocelyn Quaintance, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kwong/kwong9.html">On Certain Sums of Stirling Numbers with Binomial Coefficients</a>, J. Integer Sequences, 18 (2015), Article 15.9.6.
%H A051577 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A051577 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A051577 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.
%H A051577 Michael Wallner, <a href="https://arxiv.org/abs/1706.07163">A bijection of plane increasing trees with relaxed binary trees of right height at most one</a>, arXiv:1706.07163 [math.CO], 2017, p. 10.
%H A051577 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>.
%H A051577 Wikipedia, <a href="https://en.wikipedia.org/wiki/Double_factorial">Double factorial</a>.
%F A051577 a(n) = (2*n + 3)!!/3!!.
%F A051577 E.g.f.: 1/(1 - 2*x)^(5/2).
%F A051577 a(n) ~ (4/3) * sqrt(2) * n^2 * 2^n * e^(-n) * n^n *{1 + (47/24)*n^(-1) + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
%F A051577 Ramanujan polynomials -psi_n(n, x) evaluated at 0. - _Ralf Stephan_, Apr 16 2004
%F A051577 a(n) = 2^(2 + n) * Gamma(n + 5/2)/(3 * sqrt(Pi)). - _Gerson Washiski Barbosa_, May 05 2010
%F A051577 From _Peter Bala_, May 26 2017: (Start)
%F A051577 D-finite with recurrence a(n+1) = (2*n + 5)*a(n) with a(0) = 1.
%F A051577 O.g.f. A(x) satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 5*x)*A(x) - 1 with A(0) = 1.
%F A051577 G.f. as an S-fraction: A(x) = 1/(1 - 5*x/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 9*x/(1 - 6*x/(1 - ... - (2*n+3)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes, 1982).
%F A051577 Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 5*x/(1 - 7*x/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - ... - (2*n + 5)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
%F A051577 From _Amiram Eldar_, Dec 11 2022: (Start)
%F A051577 Sum_{n>=0} 1/a(n) = 3*(sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1), where erf is the error function.
%F A051577 Sum_{n>=0} (-1)^n/a(n) = 3*(1 - sqrt(Pi/(2*e)) * erfi(1/sqrt(2))), where erfi is the imaginary error function. (End)
%F A051577 a(n) = Sum_{k=1..n+1} (-1)^k*binomial(2*n + 1, n + k)*Stirling1(n + k + 1, k). - _Detlef Meya_, Jan 17 2024
%p A051577 seq( doublefactorial(2*n+3)/3,n=0..10) ; # _R. J. Mathar_, Sep 29 2013
%t A051577 Table[(2*n + 3)!!/3!!, {n, 0, 25}] (* _G. C. Greubel_, Jan 22 2017 *)
%t A051577 a[n_] := Sum[(-1)^k*Binomial[2*n + 1, n + k]*StirlingS1[n + k + 1 ,k], {k , 1, n + 1}]; Flatten[Table[a[n], {n, 0, 18}]] (* _Detlef Meya_, Jan 17 2024 *)
%o A051577 (PARI) vector(26, n, (2*n+2)!/(6*2^n*(n+1)!) ) \\ _G. C. Greubel_, Nov 12 2019
%o A051577 (Magma) F:=Factorial; [F(2*n+4)/(12*2^n*F(n+2)): n in [0..25]]; // _G. C. Greubel_, Nov 12 2019
%o A051577 (Sage) f=factorial; [f(2*n+4)/(12*2^n*f(n+2)) for n in (0..25)] # _G. C. Greubel_, Nov 12 2019
%o A051577 (GAP) F:=Factorial;; List([0..25], n-> F(2*n+4)/(12*2^n*F(n+2)) ); # _G. C. Greubel_, Nov 12 2019
%Y A051577 Cf. A000165, A001147, A002866(n+1) (m=0..2 rows of A(3; m,n)).
%Y A051577 Cf. A051578, A051579, A051580, A051581, A051582, A051583, A178647.
%K A051577 easy,nonn
%O A051577 0,2
%A A051577 _Wolfdieter Lang_