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Third column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).
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%I #28 Aug 09 2022 02:29:31

%S 1,18,252,3360,45360,635040,9313920,143700480,2335132800,39956716800,

%T 719220902400,13599813427200,269729632972800,5602076992512000,

%U 121645100408832000,2757288942600192000,65140951268929536000,1601701037083090944000,40932359836567879680000

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

%H Indranil Ghosh, <a href="/A062141/b062141.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+2)!*binomial(n+5, 5)/2!.

%F E.g.f.: (1 + 10*x + 10*x^2)/(1-x)^8.

%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,-3), (n >= 5). - _Milan Janjic_, Mar 01 2009

%e a(2) = (2+2)! * binomial(2+5,5) / 2! = (24*21)/2 = 252. - _Indranil Ghosh_, Feb 24 2017

%p a:= n->(n+2)!*binomial(n+5, 5)/2!: seq(a(n), n=0..20); # _Zerinvary Lajos_, Apr 29 2007

%t Table[(n+2)!*Binomial[n+5,5]/2!,{n,0,15}] (* _Indranil Ghosh_, Feb 24 2017 *)

%o (Sage) [binomial(n,5)*factorial (n-3)/2 for n in range(5, 21)] # _Zerinvary Lajos_, Jul 07 2009

%o (PARI) a(n)=(n+2)!*binomial(n+5,5)/2! \\ _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 A062141(n): return f(n+2)*C(n+5,5)/f(2) # _Indranil Ghosh_, Feb 24 2017

%o (PARI) x='x+O('x^30); Vec(serlaplace((1+10*x+10*x^2)/(1-x)^8)) \\ _G. C. Greubel_, May 11 2018

%o (Magma) [Factorial(n+2)*Binomial(n+5,5)/2: n in [0..20]]; // _G. C. Greubel_, May 11 2018

%Y Cf. A061206, A062137.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 19 2001