A051622 - OEIS

%I #23 Sep 08 2022 08:44:59

%S 1,14,252,5544,144144,4324320,147026880,5587021440,234654900480,

%T 10794125422080,539706271104000,29144138639616000,1690360041097728000,

%U 104802322548059136000,6916953288171902976000,484186730172033208320000

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

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

%C From _Zerinvary Lajos_, Feb 15 2008: (Start)

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

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

%H G. C. Greubel, <a href="/A051622/b051622.txt">Table of n, a(n) for n = 0..363</a>

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

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

%F 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

%p seq(mul((n+3+k), k=1..n+3)/120, n=0..18); # _Zerinvary Lajos_, Feb 15 2008

%t s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 4}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008 *)

%t With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(7/2), {x, 0, nn}], x]*Range[0, nn]!] (* _G. C. Greubel_, Aug 15 2018 *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(14/4))) \\ _G. C. Greubel_, Aug 15 2018

%o (Magma) m:=30; R<x>:=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

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

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_