439 
v 
mn 4 A(k, i) = A(k +1,» +1) ASOD): 2, 
i=k 
n 
i «> AK, 2) SE—1,4— 1) = 0 mee An rad 
iu 
Inversely, since 
eT Y= e(¢—1) (29 —2) Eo Ne if (x — n) op); Seem sen ee 
if we set 
(9) i Ne a|B(o, n)x” — BCI, DE ARs ek aby Bik, nya * 4 ee: 
ee oko + (—1)" Bin, n)] 
int Sevag emtnulates (Oy 7) = Ie — 0. 2 eek) = the sum of the 
products of the numbers 1, 2, 3,..... n, taken k at a time; in particular 
cen) ee — yt S(k, k) and B(k, n) = 0if k>n. For convenience define 
B(p, n) = 0, if p is a negative integer. 
If we multiply both sides of 
Phones = Bin ——1)e 4 CK) Ba = ha 19] 
os we obtain the recursion formula 
by « — n, and equate the coefficients of x” — 
(10) Bik, n) = Bik, n—1) +n B(k — 1, n —1) 
by means of which may be constructed 
A TABLE OF VALUES OF B(k, n) 
k=0 | kal] k=2|k=3 | b=4] b=5| b=-6 | k=7 | b=8 
n= 0 1 | | 
n=1 1 | 1 | 
n= 2 1 3 2 | 
n= 3 BN) | Bich itd | a6 | | | 
n= 4 te tO Bl 85) | 50) Bed | | 
n= 5 i 1) GG RPA Soe is| oo ers | 
n = 6 1 | 21 | 175°) 735 | 1624| 1764] 720 
n=T7 1 | 28 | 322 | 1960 | 6769 |13132| 13068) 5040 | 
n= 8 1 | 36 | 546 | 4536 |22449 |67284 118124 | 109584 |40320 
Multiply any entry by the number (n+1) of the next row, and add to the entry on its right. 
nt+k ,, 
(11) B(k,k+n) => | Bkt+n—i,k+n—1) Rone On ae a 
= 
