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A062143 Fifth column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

%I A062143 #24 Aug 09 2022 02:29:21
%S A062143 1,40,1080,25200,554400,11975040,259459200,5708102400,128432304000,
%T A062143 2968213248000,70643475302400,1733976211968000,43927397369856000,
%U A062143 1148870392750080000,31019500604252160000,864410083505160192000
%N A062143 Fifth column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).
%C A062143 The coefficients of the numerator polynomials N(m,x) of the e.g.f. for column m (here m=4) give triangle A062145.
%H A062143 Indranil Ghosh, <a href="/A062143/b062143.txt">Table of n, a(n) for n = 0..400</a>
%H A062143 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A062143 a(n) = (n+4)!*binomial(n+7, 7)/4!;
%F A062143 E.g.f.: (1 + 28*x + 126*x^2 + 140*x^3 + 35*x^4)/(1-x)^12.
%F A062143 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-4) = (-1)^n*f(n,4,-8), (n>=4). - _Milan Janjic_, Mar 01 2009
%e A062143 a(3) = (3+4)! * binomial(3+7,7) / 4! = (5040 * 120) / 24 = 25200. - _Indranil Ghosh_, Feb 23 2017
%t A062143 Table[(n+4)!*Binomial[n+7,7]/4!,{n,0,15}] (* _Indranil Ghosh_, Feb 23 2017 *)
%o A062143 (PARI) a(n) = (n+4)!*binomial(n+7,7)/4! \\ _Indranil Ghosh_, Feb 23 2017
%o A062143 (Python)
%o A062143 import math
%o A062143 f=math.factorial
%o A062143 def C(n,r):return f(n)/f(r)/f(n-r)
%o A062143 def A062143(n):return f(n+4)*C(n+7,7)/f(4) # _Indranil Ghosh_, Feb 23 2017
%o A062143 (Magma) [Factorial(n+4)*Binomial(n+7,7)/Factorial(4): n in [0..20]]; // _G. C. Greubel_, May 12 2018
%Y A062143 Cf. A062137, A062142, A062145.
%K A062143 nonn,easy
%O A062143 0,2
%A A062143 _Wolfdieter Lang_, Jun 19 2001