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

1, 5, 35, 315, 3465, 45045, 675675, 11486475, 218243025, 4583103525, 105411381075, 2635284526875, 71152682225625, 2063427784543125, 63966261320836875, 2110886623587616875, 73881031825566590625, 2733598177545963853125, 106610328924292590271875

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..250

H. W. Gould, Harris Kwong, and Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), Article 15.9.6.

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, p. 10.

Eric Weisstein's World of Mathematics, Double Factorial.

Wikipedia, Double factorial.

Formula

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

E.g.f.: 1/(1 - 2*x)^(5/2).

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

Ramanujan polynomials -psi_n(n, x) evaluated at 0. - Ralf Stephan, Apr 16 2004

a(n) = 2^(2 + n) * Gamma(n + 5/2)/(3 * sqrt(Pi)). - Gerson Washiski Barbosa, May 05 2010

From Peter Bala, May 26 2017: (Start)

D-finite with recurrence a(n+1) = (2*n + 5)*a(n) with a(0) = 1.

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.

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).

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)

From Amiram Eldar, Dec 11 2022: (Start)

Sum_{n>=0} 1/a(n) = 3*(sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1), where erf is the error function.

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)

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

Maple

seq( doublefactorial(2*n+3)/3, n=0..10) ; # R. J. Mathar, Sep 29 2013

Mathematica

Table[(2*n + 3)!!/3!!, {n, 0, 25}] (* G. C. Greubel, Jan 22 2017 *)
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 *)

Prog

(PARI) vector(26, n, (2*n+2)!/(6*2^n*(n+1)!) ) \\ G. C. Greubel, Nov 12 2019
(Magma) F:=Factorial; [F(2*n+4)/(12*2^n*F(n+2)): n in [0..25]]; // G. C. Greubel, Nov 12 2019
(Sage) f=factorial; [f(2*n+4)/(12*2^n*f(n+2)) for n in (0..25)] # G. C. Greubel, Nov 12 2019
(GAP) F:=Factorial;; List([0..25], n-> F(2*n+4)/(12*2^n*F(n+2)) ); # G. C. Greubel, Nov 12 2019

Crossrefs

Cf. A000165, A001147, A002866(n+1) (m=0..2 rows of A(3; m,n)).

Cf. A051578, A051579, A051580, A051581, A051582, A051583, A178647.

Sequence in context: A015683 A000357 A102147 * A124564 A260952 A349515

Adjacent sequences: A051574 A051575 A051576 * A051578 A051579 A051580

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved