A062148 - OEIS

A062148

Second (unsigned) column sequence of triangle A062138 (generalized a=5 Laguerre).

1, 14, 168, 2016, 25200, 332640, 4656960, 69189120, 1089728640, 18162144000, 319653734400, 5928123801600, 115598414131200, 2365321396838400, 50685458503680000, 1135354270482432000, 26538906072526848000, 646300418472124416000, 16372943934627151872000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..400

Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.

Index entries for sequences related to Laguerre polynomials.

FORMULA

E.g.f.: (1+6*x)/(1-x)^8.

a(n) = A062138(n+1, 1) = (n+1)!*binomial(n+6, 6).

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-1) = (-1)^(n-1) * f(n,1,-7), (n>=1). - Milan Janjic, Mar 01 2009

Assuming offset 1: a(n) = n!*binomial(-n,6). - Peter Luschny, Apr 29 2016

From Amiram Eldar, Sep 24 2022: (Start)

Sum_{n>=0} 1/a(n) = 5477/10 - 204*e - 6*gamma + 6*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) = 403/10 - 120/e + 6*gamma - 6*Ei(-1), where -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

EXAMPLE

a(3) = (3+1)! * binomial(3+6,6) = 24 * 84 = 2016. - Indranil Ghosh, Feb 24 2017

MATHEMATICA

Table[Sum[n!/6!, {i, 6, n}], {n, 6, 21}] (* Zerinvary Lajos, Jul 12 2009 *)

PROG

(PARI) a(n)=(n+1)!*binomial(n+6, 6) \\ Indranil Ghosh, Feb 24 2017

(Python)

import math

f=math.factorial

def C(n, r):return f(n)/f(r)/f(n-r)

def A062148(n): return f(n+1)*C(n+6, 6) # Indranil Ghosh, Feb 24 2017