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a(n) = (5*n+9)(!^5)/9(!^5), related to A034301 ((5*n+2)(!^5) quintic, or 5-factorials).
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%I #14 Sep 08 2022 08:44:59

%S 1,14,266,6384,185136,6294624,245490336,10801574784,529277164416,

%T 28580966878464,1686277045829376,107921730933080064,

%U 7446599434382524416,551048358144306806784,43532820293400237735936

%N a(n) = (5*n+9)(!^5)/9(!^5), related to A034301 ((5*n+2)(!^5) quintic, or 5-factorials).

%C Row m=9 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

%H G. C. Greubel, <a href="/A051690/b051690.txt">Table of n, a(n) for n = 0..352</a>

%F a(n) = ((5*n+9)(!^5))/9(!^5) = A034301(n+2)/9.

%F E.g.f.: 1/(1-5*x)^(14/5).

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

%t With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(14/5), {x, 0, nn}], x]*Range[0, nn]!] (* _G. C. Greubel_, Aug 15 2018 *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(14/5))) \\ _G. C. Greubel_, Aug 15 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^14/5))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 15 2018

%Y Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_