A096747 Triangle read by rows: T(n,1) = 1, T(n,k) = T(n-1,k)+(n-1)*T(n-1,k-1) for 1<=k<=n+1.

1, 1, 1, 1, 2, 2, 1, 4, 6, 6, 1, 7, 18, 24, 24, 1, 11, 46, 96, 120, 120, 1, 16, 101, 326, 600, 720, 720, 1, 22, 197, 932, 2556, 4320, 5040, 5040, 1, 29, 351, 2311, 9080, 22212, 35280, 40320, 40320, 1, 37, 583, 5119, 27568, 94852, 212976, 322560, 362880

Offset: 0

Author

Thomas J Engelsma (tom(AT)opertech.com), Dec 05 2004

Keywords

• nonn
• easy
• tabl

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

Examples

			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
		

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.

Crossrefs

Cf. A081046, A109822.

Programs

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 *)

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

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

Extensions

More terms from Emeric Deutsch, Jul 03 2005