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Sixth (unsigned) column of triangle A062138 (generalized a=5 Laguerre).
3

%I #23 Aug 09 2022 02:29:11

%S 1,66,2772,96096,3027024,90810720,2663781120,77630192640,

%T 2270683134720,67111301537280,2013339046118400,61498356317798400,

%U 1916698771904716800,61039483966811750400,1988143192061868441600

%N Sixth (unsigned) column of triangle A062138 (generalized a=5 Laguerre).

%H Indranil Ghosh, <a href="/A062152/b062152.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) = A062138(n+5, 5).

%F a(n) = (n+5)!*binomial(n+10, 10)/5!.

%F E.g.f.: N(5;5, x)/(1-x)^16 with N(5;5, x) := Sum_{k=0..5} A062190(5, k)* x^k = 1 + 50*x + 450*x^2 + 1200*x^3 + 1050*x^4 + 252*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-10) = (-1)^n*f(n,10,-6), (n>=10). - _Milan Janjic_, Mar 01 2009

%e a(2) = (2+5)! * binomial(2+10,10) / 5! = (5040 * 66) / 120 = 2772. - _Indranil Ghosh_, Feb 24 2017

%t Table[(n+5)!*Binomial[n+10,10]/5!,{n,0,14}] (* _Indranil Ghosh_, Feb 24 2017 *)

%o (PARI) a(n) = (n+5)!*binomial(n+10,10)/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 A062152(n): return f(n+5)*C(n+10, 10)/f(5) # _Indranil Ghosh_, Feb 24 2017

%o (Magma) [Factorial(n+5)*Binomial(n+10,10)/Factorial(5): n in [0..20]]; // _G. C. Greubel_, May 11 2018

%Y Cf. A062138, A062151.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 19 2001