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ROW AND RISING DIAGONAL SUMS FOR A TYPE OF PASCAL TRIANGLE
STEPHEN W. SMITH and DEAN B. PRIEST
Harding College, Searcy, Arkansas
As has been noted by Hoggatt [ 1 ] , Pascal's Triangle can be thought of as having been generated by column
generators. This provides insight into the row sums and rising diagonal sums of this triangle. Let {ai}o)7=0 ^e"
note a real number sequence and consider the following array:
a
a
a
a
0,0
l,0
0,2
a
a
a
l,2
a
a
2,l
a
a
a
n,l
a
l,l
2,0
a
a
0,l
n, 0
0,3
l,3
2,2
2,3
•"
a
-
a
l,m
'"
"'
a
2,m
"'
0,m
a
n,2
'"
a
n,3
n,m
which has the Pascal-like property [3]
„
a{
j
=
'f
if
[ai-i,j-i+ai-i,j
0
x
i>i>1,
j>i>0.
Under these conditions, it follows readily that
i-l
a
=
ij
E
a
Kj-l
k=0
for all / and /such that / > / > 1. For the following assume that f(x) is the generating function for the sequence
{ai,o)?=l1. The generating function for the kth column (k = 0,1,2, •-) in the above array is
Theorem
gkM
Proof.
f(x)[x/(1-x)]h,
=
Let
fix) = a0i0
+ a
i,ox
+ a
2yOx2
+
-
denote the generating function for the zeroth column {ai}o)T=i • Suppose that
f(x)[x/(1-x)]m
(a^mix1
= X
i=Q
for some positive integer m. Then by the comment preceding Theorem 1
oo
f(x)[x/(1-x)]m+l
= f(x)[x/(1-x)]m[x/(l-X>]
= Y,
/
\
i—1
E
a
km
i=0 \k=0
and the proof is complete by induction on m.
Theorem
2. The generating function for the row sums of the above array is
[f(x)(1-x)]/(1-2x).
359
j*1
J
oo
= E
i=0
fym+l)**
360
Proof.
ROW Al\lD RISING DIAGONAL SUMS FOR A TYPE OF PASCAL TRIANGLE
Since#£(x) = f(x)[x/(1 ~ xj]
, the generating function for the row sums is
GM = E ffkM = fix) £
k=o
Theorem
[DEC.
7
f 7 f ~ ) = Hx)
k=o
w
x
'
= fix)
| / -
7l-2x
73. The generating function for the rising diagonal sums of the above array is
[f(x)(1-x)]/(1-x-x2).
Proof.
Consider the new array:
0,0
0
30,1
0
0
0
0
0
0
31,0
32,0
31,1
30,2
0
0
a
32,1
31,2
30,3
a
3,0
34,0
33,1
32,2
f(x)
xgi(x)
x2g2(x)
0
a
31,3
x3g3(x)
o,4
x4g4(x)
•••
k
Note that the column generator for the k column (k = 0, 1,2, —) \sx g^(x). Furthermore the row sums of
this array are the rising diagonal sums of the original array. Thus the generating function for the rising diagonal
sums in the original array is
k
DM = £ xkgk(x) = fix) £
k=o ' '7- " '
k=o
flx)
1 - x
1-x-x
Now if f(x) = 1/(1 - x), one has the usual Pascal Triangle and some of results in [ 1 ] . Moreover, if
fix) = x[(1 - x)/(1 - 3x +x2)]
+ 1 = 11- 2x)/ll
-
3x+x2),
then the theorems above can be used to answer Problem H-183 [4] in this Quarterly. Indeed, since
H(x)
a+
=
(h-ap)x
1 - px + qx2
is the generating function for the generalized Fibonacci sequence^ = wn(a,b;p,q)
[ 2 ] , and
wi + fwj - fp - 2q)wi ]x
Olx)
1-lp2
-2q)x + q2x2
and
E(x)
-
w
O + l^2~
(P2 ~
2
2q)w0]x
+ q2x2
1-lp -2q)x
are the generating functions for the sequences [w2k+i}7=,
and {w2k}°L=lt respectively, then questions similar
to H-183 can be answered readily by considering the generating functionsxH(x) + 1,xO(x)+ 1, andxE(x)+ 1.
In particular, if one considers the sequence {a^o^g where aoyo= 1 and a^Q = L.2i~i ( f o r / = 1,2,3, •-), then
the array is
1
1
4
11
29
1
2
6
17
1
3
9
1
4
1
1977]
ROW AWD RISING DIAGONAL SUMS FOR A TYPE OF PASCAL TRIANGLE
361
the generating function for the zeroth column is
(1-2x+2x2)/(1-3x+x2),
the generating function for the row sums is
(1 - 3x + 4x2 - 2x3)/(1 -5x
+ 7x2-
2x3),
and the generating function for the rising diagonal sums is
(1 - 3x + 4x2)/(1 - 4x + 3x2 +2x3 -
x4).
REFERENCES
1. V.E. Hoggatt, Jr., "A New Angle on Pascal's Triangle," The Fibonacci Quarterly, Vol. 6, No. 4 (Dec. 1968),
pp. 221-234.
2. A. F. Horadam, "Pell Identities," The Fibonacci Quarterly, Vol. 9, No. 3 (Oct. 1971), p. 247.
3. J. G. Kemeny, H. Mirkil, J. L. Snell, and G. L Thompson, Finite Mathematical Structures, Prentice-Hall,
Inc., Englewood Cliffs, N.J., 1959, p. 93.
4. Problem H-183, 77H? Fibonacci Quarterly, Vol. 9, No. 4 (1971), p. 389, by V. E. Hoggatt, Jr.
*******
In the paper Tribonacci Sequence by April Scott,
Tom Delaney and V. E. Hoggatt,Jr* in the October FQJ
the following references should be appended.
1. Mark Feinberg, Fibonacci-Tribonacci, FQJ Oct.,1963
PP71-71*
2. Trudy Tong, Some Properties of the Tribonacci Seqqixenee and The Special Lucas Sequence, Unpublished
Masters Thesis, San Jose State University, August,1970
3* C« C. Yalavigi, Properties of the Tribonacci Numbers
FQJ Oct. 1972 pp231-2V6
**••• Krishnaswami Alladi, On Tibonacci Numbers and Related
Functions, FQJ Feb., 1977 p p ^ - 1 ^
5* A. G. Shannon, Tribonacci Numbers and Pascals Pyramid,
FQJ Oct., 1977 pp 268+ 275
The term TRIBONACCI number was coined by Mark Eeinberg
in [V]above.