A109822 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, 2, 1, 4, 6, 1, 7, 18, 24, 1, 11, 46, 96, 120, 1, 16, 101, 326, 600, 720, 1, 22, 197, 932, 2556, 4320, 5040, 1, 29, 351, 2311, 9080, 22212, 35280, 40320, 1, 37, 583, 5119, 27568, 94852, 212976, 322560, 362880, 1, 46, 916, 10366, 73639, 342964, 1066644

Offset: 1

Author

Emeric Deutsch, Jul 03 2005

Keywords

• nonn
• tabl

Comments

T(n,n) = n!. Sum of row n is the signless Stirling number of the first kind s(n,2)(A000254). T(n,k) = A096747(n,k) for 1 <= k <= n.

Examples

			T(5,3) = 46 because 18 + 4*7 = 46.
Triangle begins:
Row 1:                    1
Row 2:                 1     2
Row 3:              1     4     6
Row 4:           1     7    18    24
Row 5:        1    11    46    96   120
Row 6:     1    16   101   326   600   720
Row 7:  1    22   197   932  2556  4320  5040
		

Crossrefs

Cf. A000254, A049444, A059518, A096747, A137650.

Programs

Maple

with(combinat): T:=(n,k)->add(abs(stirling1(n,n-i)),i=0..k-1): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form T:=proc(n,k) if k=1 then 1 elif k=n then n! else T(n-1,k)+(n-1)*T(n-1,k-1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form. - Emeric Deutsch, Jul 03 2005
A109822_row := proc(n) local k,i;
add(add(abs(combinat[stirling1](n, n-i)), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,n-k),k=1..n) end:
seq(print(A109822_row(n)),n=1..7); # Peter Luschny, Sep 18 2011

Mathematica

Table[Sum[Abs@ StirlingS1[n, n - i], {i, 0, k - 1}], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Aug 17 2017 *)

Formula

T(n, k) = Sum_{i=0..k-1} |stirling1(n, n-i)| for 1 <= k <= n.

From Peter Bala, Jul 08 2012: (Start)

E.g.f.: x/(1-x)*{1/(1-x*z)^(1/x) - 1/(1-x*z)} = x*z + (x + 2*x^2)*z^2/2! + (x + 4*x^2 + 6*x^3)*z^3/3! + ... Cf. the e.g.f. of A059518.

(End)