OFFSET
0,2
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.
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)
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 *)
PROG
(Sage) [binomial(n, 5)*factorial (n-4) for n in range(5, 22)] # Zerinvary Lajos, Jul 07 2009
(Magma) [Binomial(n, 5)*Factorial(n-4): n in [5..25]]; // Vincenzo Librandi, Feb 23 2014
(PARI) x='x+O('x^30); Vec(serlaplace((1+5*x)/(1-x)^7)) \\ G. C. Greubel, Feb 07 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jun 19 2001
EXTENSIONS
More terms from Vincenzo Librandi, Feb 23 2014
STATUS
approved