A062144 - OEIS

%I #20 Sep 08 2022 08:45:03

%S 1,54,1890,55440,1496880,38918880,998917920,25686460800,667847980800,

%T 17660868825600,476843458291200,13178219210956800,373382877643776000,

%U 10856825211488256000,324153781314435072000

%N Sixth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

%H Indranil Ghosh, <a href="/A062144/b062144.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>

%F a(n) = (n+5)!*binomial(n+8, 8)/5!.

%F E.g.f.: N(3;5, x)/(1-x)^14 with N(3;5, x) := Sum_{k=0..5} A062145(5, k) *x^k = 1 +40*x +280*x^2 +560*x^3 +350*x^4 +56*x^5.

%F If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n-5) = (-1)^(n-1)*f(n,5, -9), (n>=5). - _Milan Janjic_, Mar 01 2009

%e a(2) = (2+5)! * binomial(2+8,8)/ 5! = (5040 * 45) / 120 = 1890. - _Indranil Ghosh_, Feb 24 2017

%t Table[(n+5)!*Binomial[n+8,8]/5!,{n,0,14}] (* _Indranil Ghosh_, Feb 24 2017 *)

%o (PARI) a(n)=(n+5)!*binomial(n+8, 8)/5! \\ _Indranil Ghosh_, Feb 24 2017

%o (Python)

%o import math

%o f=math.factorial

%o def C(n, r):return f(n)/f(r)/f(n-r)

%o def A062144(n): return f(n+5)*C(n+8, 8)/f(5) # _Indranil Ghosh_, Feb 24 2017

%o (Magma) [Factorial(n+5)*Binomial(n+8, 8)/Factorial(5): n in [0..20]]; // _G. C. Greubel_, May 11 2018

%Y Cf. A062143.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 19 2001