OFFSET
0,2
COMMENTS
The coefficients of the numerator polynomials N(m,x) of the e.g.f. for column m (here m=4) give triangle A062145.
LINKS
FORMULA
a(n) = (n+4)!*binomial(n+7, 7)/4!.
E.g.f.: (1 + 28*x + 126*x^2 + 140*x^3 + 35*x^4)/(1-x)^12.
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-4) = (-1)^n*f(n,4,-8), (n >= 4). - Milan Janjic, Mar 01 2009
From Amiram Eldar, Nov 04 2025: (Start)
a(n) ~ sqrt(Pi/2) * n^(n+23/2) / (60480 * exp(n)).
Sum_{n>=0} 1/a(n) = 1064*e + 1316*(gamma - Ei(1)) - 17353/15, where e = A001113, Ei(1) = A091725, and gamma = A001620.
Sum_{n>=0} (-1)^n/a(n) = 6412*(gamma - Ei(-1)) - 3920/e - 54971/15, where Ei(-1) = -A099285. (End)
EXAMPLE
a(3) = (3+4)! * binomial(3+7,7) / 4! = (5040 * 120) / 24 = 25200. - Indranil Ghosh, Feb 23 2017
MATHEMATICA
Table[(n+4)!*Binomial[n+7, 7]/4!, {n, 0, 15}] (* Indranil Ghosh, Feb 23 2017 *)
PROG
(PARI) a(n) = (n+4)!*binomial(n+7, 7)/4! \\ Indranil Ghosh, Feb 23 2017
(Python)
import math
f=math.factorial
def C(n, r):return f(n)/f(r)/f(n-r)
def A062143(n):return f(n+4)*C(n+7, 7)/f(4) # Indranil Ghosh, Feb 23 2017
(Magma) [Factorial(n+4)*Binomial(n+7, 7)/Factorial(4): n in [0..20]]; // G. C. Greubel, May 12 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jun 19 2001
STATUS
approved