A051622 a(n) = (4*n+10)(!^4)/10(!^4), related to A000407 ((4*n+2)(!^4) quartic, or 4-factorials).

1, 14, 252, 5544, 144144, 4324320, 147026880, 5587021440, 234654900480, 10794125422080, 539706271104000, 29144138639616000, 1690360041097728000, 104802322548059136000, 6916953288171902976000, 484186730172033208320000

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Row m=10 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

From Zerinvary Lajos, Feb 15 2008: (Start)

a(n) = A001813(n+3)/120.

a(n) = A051618(n+1)/10. (End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..363

Crossrefs

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051621 (rows m=0..9).

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(14/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

Maple

seq(mul((n+3+k), k=1..n+3)/120, n=0..18); # Zerinvary Lajos, Feb 15 2008

Mathematica

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(7/2), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(14/4))) \\ G. C. Greubel, Aug 15 2018

Formula

a(n) = ((4*n+10)(!^4))/10(!^4) = A000407(n+2)/(6*10).

E.g.f.: 1/(1-4*x)^(7/2).

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+7)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013