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A062199 Second (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).

%I A062199 #46 May 09 2024 13:26:06
%S A062199 1,12,126,1344,15120,181440,2328480,31933440,467026560,7264857600,
%T A062199 119870150400,2092278988800,38532804710400,746943599001600,
%U A062199 15205637551104000,324386934423552000,7237883474325504000,168600109166641152000,4093235983656787968000
%N A062199 Second (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).
%C A062199 a(n) is the total number of ascending runs of length 5 over all permutations of {1,2,...,n+5}.  a(1) = 12 because we have: [1,2,3,4,6,5], [1,2,3,5,6,4], [1,2,4,5,6,3], [1,3,4,5,6,2], [2,1,3,4,5,6], [2,3,4,5,6,1], [3,1,2,4,5,6], [4,1,2,3,5,6], [5,1,2,3,4,6], [6,1,2,3,4,5], and [1,2,3,4,5,6] which has two runs of length 5. - _Geoffrey Critzer_, Feb 21 2014
%H A062199 Vincenzo Librandi, <a href="/A062199/b062199.txt">Table of n, a(n) for n = 0..200</a>
%H A062199 Milan Janjić, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A062199 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>.
%F A062199 E.g.f.: (1+5*x)/(1-x)^7.
%F A062199 a(n) = A062140(n+1, 1) = (n+1)!*binomial(n+5, 5).
%F A062199 If we define f(n,i,x)= Sum(Sum(binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n-1)=(-1)^(n-1)*f(n,1,-6), (n>=1). [_Milan Janjic_, Mar 01 2009]
%F A062199 a(n) = Sum_{k>0} k * A264781(n+5,k). - _Alois P. Heinz_, Nov 24 2015
%F A062199 Assuming offset 1: a(n) = -n!*binomial(-n,5). - _Peter Luschny_, Apr 29 2016
%F A062199 From _Amiram Eldar_, Sep 24 2022: (Start)
%F A062199 Sum_{n>=0} 1/a(n) = 1565/12 - 50*e - 5*gamma + 5*Ei(1), where gamma is Euler's constant (A001620) and Ei(1) is the exponential integral at 1 (A091725).
%F A062199 Sum_{n>=0} (-1)^n/a(n) = -125/12 + 20/e + 5*gamma - 5*Ei(-1), where -Ei(-1) is the negated exponential integral at -1 (A099285). (End)
%t A062199 Table[Sum[n!/5!, {i, 5, n}], {n, 5, 21}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%t A062199 With[{nn=20},CoefficientList[Series[(1+5x)/(1-x)^7,{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Nov 10 2016 *)
%o A062199 (Sage) [binomial(n,5)*factorial (n-4) for n in range(5, 22)] # _Zerinvary Lajos_, Jul 07 2009
%o A062199 (Magma) [Binomial(n, 5)*Factorial(n-4): n in [5..25]]; // _Vincenzo Librandi_, Feb 23 2014
%o A062199 (PARI) x='x+O('x^30); Vec(serlaplace((1+5*x)/(1-x)^7)) \\ _G. C. Greubel_, Feb 07 2018
%Y A062199 Cf. A001720 (first column of A062140), A264781.
%Y A062199 Cf. A001620, A091725, A099285.
%K A062199 nonn,easy
%O A062199 0,2
%A A062199 _Wolfdieter Lang_, Jun 19 2001
%E A062199 More terms from _Vincenzo Librandi_, Feb 23 2014