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%I #26 May 06 2022 07:35:26
%S 1,24,420,6720,105840,1693440,27941760,479001600,8562153600,
%T 159826867200,3116623910400,63465795993600,1348648164864000,
%U 29877743960064000,689322235650048000,16543733655601152000,412559358036553728000,10678006913887272960000,286526518855975157760000
%N Fourth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).
%H Harry J. Smith, <a href="/A062193/b062193.txt">Table of n, a(n) for n = 0..100</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>.
%F E.g.f.: (1+15*x+30*x^2+10*x^3)/(1-x)^9.
%F a(n) = A062139(n+3, 3).
%F a(n) = (n+3)!*binomial(n+5, 5)/3!.
%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-3)=(-1)^(n-1)*f(n,3,-6), (n>=3). - _Milan Janjic_, Mar 01 2009
%F From _Amiram Eldar_, May 06 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 75*(Ei(1) - gamma) - 30*e - 65/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.
%F Sum_{n>=0} (-1)^n/a(n) = 315*(gamma - Ei(-1)) - 180/e - 735/4, where Ei(-1) = -A099285. (End)
%t With[{nn=20},CoefficientList[Series[(1+15*x+30*x^2+10*x^3)/(1-x)^9, {x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 02 2018 *)
%o (Sage) [binomial(n,5)*factorial (n-2)/6 for n in range(5, 21)] # _Zerinvary Lajos_, Jul 07 2009
%o (PARI) { f=2; for (n=0, 100, f*=n + 3; write("b062193.txt", n, " ", f*binomial(n + 5, 5)/6) ) } \\ _Harry J. Smith_, Aug 02 2009
%o (PARI) x='x+O('x^30); Vec(serlaplace((1+15*x+30*x^2+10*x^3)/(1-x)^9)) \\ _G. C. Greubel_, May 11 2018
%o (Magma) [Factorial(n+3)*binomial(n+5, 5)/Factorial(3): n in [0..30]]; // _G. C. Greubel_, May 11 2018
%Y Cf. A001710, A005990, A005461, A062139.
%Y Cf. A001113, A001620, A091725, A099285.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jun 19 2001
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