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A062194 Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).

%I A062194 #33 May 06 2022 07:38:01
%S A062194 1,35,840,17640,352800,6985440,139708800,2854051200,59935075200,
%T A062194 1298593296000,29088489830400,674324082432000,16183777978368000,
%U A062194 402104637462528000,10339833534750720000,275039572024369152000
%N A062194 Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).
%H A062194 Harry J. Smith, <a href="/A062194/b062194.txt">Table of n, a(n) for n = 0..100</a>
%H A062194 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>.
%F A062194 E.g.f.: (1 + 24*x + 90*x^2 + 80*x^3 + 15*x^4)/(1-x)^11.
%F A062194 a(n) = A062139(n+4, 4).
%F A062194 a(n) = (n+4)!*binomial(n+6, 6)/4!.
%F A062194 If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..n} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-4) = (-1)^n*f(n,4,-7), (n >= 4). - _Milan Janjic_, Mar 01 2009
%F A062194 From _Amiram Eldar_, May 06 2022: (Start)
%F A062194 Sum_{n>=0} 1/a(n) = 336*(gamma - Ei(1)) - 96*e + 3524/5, where gamma = A001620, Ei(1) = A091725, and e = A001113.
%F A062194 Sum_{n>=0} (-1)^n/a(n) = 3264*(gamma - Ei(-1)) - 1920/e - 9464/5, where Ei(-1) = -A099285. (End)
%t A062194 Table[(n+4)!*Binomial[n+6,6]/4!, {n, 0, 20}] (* _G. C. Greubel_, May 12 2018 *)
%o A062194 (Sage) [binomial(n,6)*factorial (n-2)/factorial (4) for n in range(6, 22)] # _Zerinvary Lajos_, Jul 07 2009
%o A062194 (PARI) { f=6; for (n=0, 100, f*=n + 4; write("b062194.txt", n, " ", f*binomial(n + 6, 6)/24) ) } \\ _Harry J. Smith_, Aug 02 2009
%o A062194 (Magma) [Factorial(n+4)*Binomial(n+6, 6)/Factorial(4): n in [0..20]]; // _G. C. Greubel_, May 12 2018
%o A062194 (GAP) List([0..15],n->Factorial(n+4)*Binomial(n+6,6)/Factorial(4)); # _Muniru A Asiru_, Jul 01 2018
%Y A062194 Cf. A001710, A005461, A005990, A062193.
%Y A062194 Cf. A001113, A001620, A091725, A099285.
%K A062194 nonn,easy
%O A062194 0,2
%A A062194 _Wolfdieter Lang_, Jun 19 2001