OFFSET
0,5
COMMENTS
Note: rows continue as factorials - stopped at second factorial for clarity.
T(n,n) = T(n,n+1) = n!. Sum of row n = n! + s(n,2), where s(n,2) are signless Stirling numbers of the first kind (A081046). T(n,k) = A109822(n,k) for 1<=k<=n (i.e. triangle without the last column is A109822). - Emeric Deutsch, Jul 03 2005
Sum(k=0..n-1, T(n,k))/T(n,n-1) are for n>=1 the harmonic numbers A001008(n)/A002805(n). - Peter Luschny, Sep 15 2014
LINKS
Robert Israel, Table of n, a(n) for n = 0..10152 (rows 0..141, flattened)
R. P. Stanley, Ordering events in Minkowski space, arXiv:math/0501256 [math.CO], 2005.
FORMULA
T(n+1, i) = n*T(n, i-1)+T(n, i)
T(n, k) = sum(|stirling1(n, n-i)|, i=0..k-1) for 1<=k<=n. - Emeric Deutsch, Jul 03 2005
E.g.f. as triangle: g(x,y) = Sum_{n>=0} Sum_{1<=k<=n+1} T(n,k) x^n y^k/n! where
g(x,y) = -y^2/((y-1)*(x*y-1)) - (1-x*y)^(-1/y)*(-y+y^2/(y-1)). - Robert Israel, Nov 28 2016
EXAMPLE
Triangle begins:
*0.........................1
*1......................1.....1
*2...................1.....2.....2
*3................1.....4.....6.....6
*4.............1.....7....18....24....24
*5..........1....11....46....96...120...120
*6.......1....16...101...326...600...720...720
*7....1....22...197...932..2556..4320..5040..5040
T(5,3)=46 because 4*7+18=46
MAPLE
T:=proc(n, k) if k=1 then 1 elif k=n+1 then n! else T(n-1, k)+(n-1)*T(n-1, k-1) fi end: for n from 0 to 11 do seq(T(n, k), k=1..n+1) od; # yields sequence in triangular form
with(combinat): T:=(n, k)->sum(abs(stirling1(n, n-i)), i=0..k-1): for n from 0 to 11 do seq(T(n, k), k=1..n+1) od; # yields sequence in triangular form; Emeric Deutsch, Jul 03 2005
MATHEMATICA
T[n_, k_] := Sum[Abs[StirlingS1[n, n - i]], {i, 0, k}]; T[0, 0] := 1;
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 08 2016 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if n == 0: return 1
if k < 0: return 0
return T(n-1, k)+(n-1)*T(n-1, k-1)
for n in range(9): print([T(n, k) for k in (0..n)]) # Peter Luschny, Sep 15 2014
CROSSREFS
KEYWORD
AUTHOR
Thomas J Engelsma (tom(AT)opertech.com), Dec 05 2004
EXTENSIONS
More terms from Emeric Deutsch, Jul 03 2005
STATUS
approved