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

1, 12, 126, 1344, 15120, 181440, 2328480, 31933440, 467026560, 7264857600, 119870150400, 2092278988800, 38532804710400, 746943599001600, 15205637551104000, 324386934423552000, 7237883474325504000, 168600109166641152000, 4093235983656787968000

Offset: 0

Author

Wolfdieter Lang, Jun 19 2001

Keywords

• nonn
• easy

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.
• Index entries for sequences related to Laguerre polynomials.

Crossrefs

Cf. A001720 (first column of A062140), A264781.

Cf. A001620, A091725, A099285.

Programs

Magma

[Binomial(n, 5)*Factorial(n-4): n in [5..25]]; // Vincenzo Librandi, Feb 23 2014

Mathematica

Table[Sum[n!/5!, {i, 5, n}], {n, 5, 21}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{nn=20},CoefficientList[Series[(1+5x)/(1-x)^7,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Nov 10 2016 *)

PARI

x='x+O('x^30); Vec(serlaplace((1+5*x)/(1-x)^7)) \\ G. C. Greubel, Feb 07 2018

Sage

[binomial(n,5)*factorial (n-4) for n in range(5, 22)] # Zerinvary Lajos, Jul 07 2009

Formula

E.g.f.: (1+5*x)/(1-x)^7.

a(n) = A062140(n+1, 1) = (n+1)!*binomial(n+5, 5).

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]

a(n) = Sum_{k>0} k * A264781(n+5,k). - Alois P. Heinz, Nov 24 2015

Assuming offset 1: a(n) = -n!*binomial(-n,5). - Peter Luschny, Apr 29 2016

From Amiram Eldar, Sep 24 2022: (Start)

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).

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)

Extensions

More terms from Vincenzo Librandi, Feb 23 2014