A051582 a(n) = (2*n+8)!!/8!!, related to A000165 (even double factorials).

1, 10, 120, 1680, 26880, 483840, 9676800, 212889600, 5109350400, 132843110400, 3719607091200, 111588212736000, 3570822807552000, 121407975456768000, 4370687116443648000, 166086110424858624000, 6643444416994344960000

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Row m=8 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..399
• A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Crossrefs

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

Cf. A051577, A051578, A051579, A051580, A051581 (rows m=0..7), A051583.

Programs

GAP

F:=Factorial;; List([0..20], n-> 2^n*F(n+4)/F(4) ); # G. C. Greubel, Nov 12 2019

Magma

F:=Factorial; [2^n*F(n+4)/F(4): n in [0..20]]; // G. C. Greubel, Nov 12 2019

Maple

seq(2^n*pochhammer(5, n), n=0..20); # G. C. Greubel, Nov 12 2019

Mathematica

(2Range[0,20]+8)!!/8!! (* Harvey P. Dale, Feb 03 2013 *)
Table[2^n*Pochhammer[5, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

PARI

vector(20, n, n--; (n+4)!*2^(n-1)/12) \\ Michel Marcus, Feb 09 2015

Sage

f=factorial; [2^n*f(n+4)/f(4) for n in (0..20)] # G. C. Greubel, Nov 12 2019

Formula

a(n) = (2*n+8)!!/8!!.

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

a(n) = (n+4)!*2^(n-1)/12. - Zerinvary Lajos, Sep 23 2006

From Peter Bala, May 26 2017: (Start)

a(n+1) = (2*n + 10)*a(n) with a(0) = 1.

O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 10*x)*A(x) - 1 with A(0) = 1.

G.f. as an S-fraction: A(x) = 1/(1 - 10*x/(1 - 2*x/(1 - 12*x/(1 - 4*x/(1 - 14*x/(1 - 6*x/(1 - ... - (2*n + 8)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).

Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 10*x/(1 - 12*x/(1 - 2*x/(1 - 14*x/(1 - 4*x/(1 - 16*x/(1 - 6*x/(1 - ... - (2*n + 10)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)

From Amiram Eldar, Dec 11 2022: (Start)

Sum_{n>=0} 1/a(n) = 384*sqrt(e) - 632.

Sum_{n>=0} (-1)^n/a(n) = 384/sqrt(e) - 232. (End)