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A062142 Fourth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

%I A062142 #28 Aug 09 2022 02:29:26
%S A062142 1,28,560,10080,176400,3104640,55883520,1037836800,19978358400,
%T A062142 399567168000,8310997094400,179819755315200,4045944494592000,
%U A062142 94612855873536000,2297740785500160000,57903067794604032000
%N A062142 Fourth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).
%H A062142 Indranil Ghosh, <a href="/A062142/b062142.txt">Table of n, a(n) for n = 0..400</a>
%H A062142 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A062142 a(n) = (n+3)!*binomial(n+6, 6)/3!; e.g.f.: (1 + 18*x + 45*x^2 + 20*x^3)/(1-x)^10.
%F A062142 If we define f(n,i,x) = Sum_{k=1..n} Sum_{j=1..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j), then a(n-3) = (-1)^(n-1)*f(n,3,-7), (n>=3). - _Milan Janjic_, Mar 01 2009
%e A062142 a(3) = (3+3)!*binomial(3+6,6)/3! = (720*84)/6 = 10080. - _Indranil Ghosh_, Feb 23 2017
%t A062142 Table[(n+3)!*Binomial[n+6,6]/3!,{n,0,15}] (* _Indranil Ghosh_, Feb 23 2017 *)
%o A062142 (Sage) [binomial(n,6)*factorial(n-3)/factorial(3) for n in range(6, 22)] # _Zerinvary Lajos_, Jul 07 2009
%o A062142 (PARI) a(n) =(n+3)!*binomial(n+6,6)/3! \\ _Indranil Ghosh_, Feb 23 2017
%o A062142 (Python)
%o A062142 import math
%o A062142 f=math.factorial
%o A062142 def C(n,r):
%o A062142     return f(n)/f(r)/f(n-r)
%o A062142 def A062142(n):return f(n+3)*C(n+6,6)/f(3) # _Indranil Ghosh_, Feb 23 2017
%o A062142 (Magma) [Factorial(n+3)*Binomial(n+6,6)/6: n in [0..20]]; // _G. C. Greubel_, May 12 2018
%Y A062142 Cf. A062137, A062141.
%K A062142 nonn,easy
%O A062142 0,2
%A A062142 _Wolfdieter Lang_, Jun 19 2001