SMARANDACHE STAR (STIRLING)
DERIVED SEQUENCES
Amarnath Murthy, S.E.(E&T), WLS, Oil and Natural Gas Corporation Ltd., Sabarmati, Ahmedabad,380005 INDIA.
Let b_{1}, b_{2}, b_{3}, . . . be a sequence say S_{b} the base sequence. Then the Smarandache star derived sequence S_{d} using the following star triangle {ref. [1]} is defined
1 




1 
1 



1 
3 
1 


1 
7 
6 
1 

1 
15 
25 
10 
1 
. . .
as follows
d_{1} = b_{1}
d_{2} = b_{1} + b_{2}
d_{3} = b_{1} + 3b_{2} + b_{3}
d_{4} = b_{1} + 7b_{2} + 6b_{3} + b_{4}
. . .
n
d_{n+1} = S a_{(m,r)} .b_{k+1}
k=0
where a_{(m,r)} is given by
r
a_{(m,r)} = (1/r!) S (1)^{rt} .^{r}C_{t}
.t^{m} , Ref. [1]
t=0
e.g. (1) If the base sequence S_{b} is 1, 1, 1, . . . then the derived sequence S_{d} is
1, 2, 5, 15, 52, . . . , i.e. the sequence of Bell numbers. T_{n} = B_{n}
(2) S_{b}  1, 2, 3, 4, . . . then
S_{d}  1, 3, 10, 37, . . ., we have T_{n} = B_{n+1}
B_{n} . Ref [1]
The Significance of the above transformation will be clear when we consider the inverse transformation. It is evident that the star triangle is nothing but the Stirling Numbers of the Second kind ( Ref. [2] ). Consider the inverse Transformation : Given the Smarandache Star Derived Sequence S_{d} , to retrieve the original base sequence S_{b }. We get b_{k} for k = 1, 2, 3, 4 etc. as follows ;
b_{1} = d_{1}
b_{2} = d_{1} + d_{2}
b_{3} = 2d_{1}  3d_{2} + d_{3}
b_{4} = 6d_{1} + 11d_{2}  6d_{3} + d_{4}
b_{5} = 24d_{1}  50d_{2} + 35d_{3}  10d_{4} + d_{5}
………………
we notice that the triangle of coefficients is
1 




1 
1 



2 
3 
1 


6 
11 
6 
1 

24 
50 
35 
10 
1 
Which are nothing but the Stirling numbers of the first kind.
Some of the properties are
(1) The first column numbers are (1) ^{r1}.(r1)! , where r is the row number.
More properties can be found in Ref. [2].
This provides us with a relationship between the Stirling numbers of the first kind and that of the second kind, which can be better expressed in the form of a matrix.
Let [b_{1,k}]_{1xn} be the row matrix of the base sequence.
[d_{1,k}]_{1xn} be the row matrix of the derived sequence.
[S_{j,k}]_{nxn} be a square matrix of order n in which s_{j,k }is the k^{th} number in the j^{th} row of the star triangle ( array of the Stirling numbers of the second kind , Ref. [2] ). Then we have
[T_{j,k}]_{nxn} be a square matrix of order n in which t_{j,k }is the k^{th} number in the j^{th} row of the array of the Stirling numbers of the first kind , Ref. [2] ). Then we have
[b_{1,k}]_{1xn} * [S_{j,k}]^{'}_{nxn} = [d_{1,k}]_{1xn}
[d_{1,k}]_{1xn} * [T_{j,k}]^{'}_{nxn} = [b_{1,k}]_{1xn}
Which suggests that [T_{j,k}]^{'}_{nxn} is the
transpose of the inverse of the transpose of the Matrix [S_{j,k}]^{'}_{nxn}
.
The proof of the above proposition is inherent in theorem 10.1 of ref.
[3].
Readers can try proofs by a combinatorial approach or otherwise.
REFERENCES:
[1] "Amarnath Murthy", 'Properties of the Smarandache Star Triangle' , SNJ, Vol. 11, No. 123, 2000.
[2] "V. Krishnamurthy" , 'COMBINATORICS Theory and applications' ,East West Press Private Limited, 1985.
[3] " Amarnath Murthy", 'Miscellaneous results and theorems on Smarandache Factor Partitions.', SNJ,Vol. 11,No. 123, 2000.
_{ }