Asia Pacific Journal of Multidisciplinary Research, Vol. 3, No. 4, November 2015 Part III ______
Asia Pacific Journal of Recurrence Relations and Generating Multidisciplinary Research Functions of the Sequence of Sums of Vol. 3 No. 4, 165-169 November 2015 Part III Corresponding Factorials and Triangular P-ISSN 2350-7756 Numbers E-ISSN 2350-8442 www.apjmr.com
Romer C. Castillo, M.Sc. Batangas State University, Batangas City, Philippines [email protected]
Date Received: September 26, 2015; Date Revised: October 30, 2015
Abstract – This study established some recurrence relations and exponential generating functions of the sequence of factoriangular numbers. A factoriangular number is defined as a sum of corresponding factorial and triangular number. The proofs utilize algebraic manipulations with some known results from calculus, particularly on power series and Maclaurin’s series. The recurrence relations were found by manipulating the formula defining a factoringular number while the ascertained exponential generating functions were in the closed form.
Keywords – closed formula, factorial, factoriangular number, exponential generating function, recurrence relation, sequence, triangular number
INTRODUCTION for determining as many terms of the sequence as The succession of numbers formed according to a desired [3]. For instance, given a term of a triangular fixed rule is called a sequence [1]. A function whose number Tn n( n 1)/ 2 , the succeeding terms in the domain is the set {1, 2, ..., n , ...} is called a sequence sequence of triangular numbers are determined by the function and the numbers in the range are called relation T T ( n 1). elements of the sequence [2]. For example, nn1 In the study of sequences and recurrences, it is 1, 4, 9, 16, 25, 36, ... is a sequence having the rule usually helpful to represent the sequence by power that the nth term is given by n2 , while series such as 1, 3, 6, 10, 15, 21, ... is a sequence having the rule Ax( ) axn a axax 23 ax ..., that the nth term is given by nn( 1)/ 2 . The elements n 0 1 2 3 n0 of the first example sequence are called square numbers the ordinary generating function of the sequence, or while the elements of the second example sequence are xn x x23 x called triangular numbers. If f is the function defining E( x ) a a a a a ... , n n!0 1 1! 2 2! 3 3! the sequence, then the sequence function is f() n n2 n0 the exponential generating function. Generating for the first example and f( n ) n ( n 1)/ 2 for the functions provide a very efficient way of representing second example. A sequence can be denoted by the sequences [3]. Note that a power series is a function notation {fn ( )} or by the subscript notation generalization of a polynomial function [2]. {}a . Thus, if f( n ) T n ( n 1)/ 2 then the When a recurrence for a sequence has been found, n n its generating function (GF) can be solved by using sequence of triangular numbers may also be denoted by several techniques (see [3]). For example, the Fibonacci either {}Tn or {nn ( 1)/ 2}. numbers 1, 1, 2, 3, 5, ... are generated by a relation Given the sequence a, a , a , a , ..., there is also 0 1 2 3 an a n12 a n , from which it is not difficult to obtain a systematic way of getting the nth term from the a closed form GF, preceding terms aann12, , ... . The rule for doing this is called recurrence relation, which provides a method 165 P-ISSN 2350-7756 | E-ISSN 2350-8442 | www.apjmr.com Castillo, Recurrence Relations and Generating Functions of the Sequence… ______
1 2, 5, 12, 34, 135, 741, 5068, 40356, 362925, ..., , 1xx2 which appears as sequence A101292 in [4]. as follows: The name factoriangular was introduced in [5] as a Let contraction of the terms factorial and triangular, together with some preliminary results on the nn2 Ax( ) axnn aaxax0 1 2 ... ax ... characteristics of such numbers including parity, n0 compositeness, number and sum of positive divisors, be the GF. Since a0 1, abundancy and deficiency, Zeckendorf’s A( x ) 1 a x a x2 ... a xn ... decomposition, end digits and digital roots. In [6], 12 n factoriangular numbers were represented as runsums and since also a1 1, and as difference of two triangular numbers, and the A( x ) 1 x a x2 ... a xn .... politeness, as well as trapezoidal arrangements 2 n associated with factoriangular numbers were also Since an a n12 a n it further follows that studied. Further in [7], interesting results on the n representations of factoriangular numbers as sum of two A( x ) 1 x an x triangular numbers and as sum of two squares were n2 presented. n A( x ) 1 x ( ann12 a ) x n2 METHODS This study employs basic and expository research nn method. It utilizes mathematical exposition and A( x ) 1 x ann12 x a x nn22 exploration, along with ingenuity and inventiveness to 2 3 4 arrive at a proof. It basically uses algebraic methods and A( x ) 1 x ( a1 x a 2 x a 3 x ...) manipulations but utilizing also known results from 2 3 4 (a0 x a 1 x a 2 x ...) calculus, particularly on power series and Maclaurin’s A( x ) 1 x x ( a x a x23 a x ...) series. 1 2 3 22 x ( a0 a 1 x a 2 x ...) RESULTS AND DISCUSSION A( x ) 1 x x ( A ( x ) 1) x2 A ( x ) A factoriangular number was defined in [5] as: Definition. The nth factoriangular number is given by A( x ) 1 x xA ( x ) x x2 A ( x ) the formula Ftnn n! T , where nn! 1 2 3 and A( x ) 1 xA ( x ) x2 A ( x ) Tn 123... n n ( n 1)/2 . 2 A( x ) xA ( x ) x A ( x ) 1 To find a relationship between two consecutive 2 (1x x ) A ( x ) 1 factoriangular numbers, Ftn and Ftn+1, the equations 1 defining them can be manipulated to have n! on one side of the equation as follows: Ax() 2 . 1xx Ft n! T A great deal had also been written about how nn generating functions can be used in mathematics (see Ftnn T n! [3] for some references). nn( 1) Ft n!, n 2 OBJECTIVES OF THE STUDY and This study aimed to establish some recurrence relations and exponential generating functions of the Ftnn11( n 1)! T relatively new sequence of numbers, the so-called Ftnn11 T ( n 1)! factoriangular numbers. These numbers are found by (nn 1)( 2) adding corresponding factorials and triangular numbers, Ft ( n 1) n ! n1 2 nT! n , and forming the sequence
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Ft n 2 2 n1 n!. nn21 Ftnn n Ft 1 n 12 2 Equating the two expressions that are both equal to holds true for factoriangular numbers. n! and then simplifying, Notice the restriction n ≥ 2. If n = 1 then Ft = Ft , Ft n2 n ( n 1) n-1 0 n1 Ft which is excluded in the definition of factoriangular n 1 2n 2 number. Nevertheless, if Ft000! T 1 0 1 is to Ft n2 n n 2 n1 Ft be included, Theorem 2 still holds true. Similarly, if Ft0 n 1n 2 2 is to be included, Theorem 1 still holds true for n ≥ 0. n2 2 But n cannot be equal to zero for Theorem 2 since Ft-1 Ftnn1 ( n 1) Ft . is not defined. 2 With the formula defining a factoriangular number, The above serves as proof for: the first factoriangular number is computed as Theorem 1. For n ≥ 1, the factoriangular numbers 1(1 1) Ft 1! 2 . follow the recurrence relation 1 2 n2 2 Then, subsequent factoriangular numbers may be Ft( n 1) Ft . nn1 computed by using a recurrence relation. For instance, 2 given that Ft1 = 2, then using the relation in Theorem 1, In an analogous way, another recurrence relation the next factoriangular numbers are computed as involving Ftn and Ftn-1 can be established. Consider the follows: following: 122 Ft n! T nn Ft2 2 2 5 , 2 Ftnn T n! 2 nn( 1) 22 Ft n( n 1)! Ft3 3 5 12 , n 2 2 Ft n 1 2 n (n 1)!, 32 Ft4 4 12 34 , n 2 2 and and so on. Ft( n 1)! T nn11 Using the recurrence relation, all factoriangular
Ftnn11 T ( n 1)! numbers can be generated. The sequence of nn( 1) factoriangular numbers is given by Ftn1 ( n 1)!. {Ft } 2,5,12,34,135,741,5068,40356,362925,... 2 n Equating these two and simplifying, for n ≥ 1. This sequence has an exponential generating function (EGF) stated in the following theorem: Therorem 3. The exponential generating function of Ftn Ftn n1 n ( n 1) Ftn1 for n ≥ 1 is given by the closed formula n 22 2 3 4 x 2 2 (2 5x 2 x x ) e n n n 1 Ex() 2 Ftnn n Ft 1 2(1 x ) 22 for 11 x . nn2 21 Proof: Ftnn n Ft 1 . For the sequence 2, 5, 12, 34, 135, ... , the EGF is 2 x2 x 3 x 4 The above is the proof of a theorem that is stated E( x ) 2 5 x 12 34 135 ... formally as: 2! 3! 4! Theorem 2. For n ≥ 2, the recurrence relation and each term can be decomposed into having
167 P-ISSN 2350-7756 | E-ISSN 2350-8442 | www.apjmr.com Asia Pacific Journal of Multidisciplinary Research, Vol. 3, No. 4, November 2015 Castillo, Recurrence Relations and Generating Functions of the Sequence… ______
x2 x 3 x 4 This can also be decomposed into two sets of Ex( ) 1 2x 6 24 120 ... summands to have 2! 3! 4! x2 x 3 x 4 x2 x 3 x 4 1x 2 6 24 ... 1 3x 6 10 15 ... . 2! 3! 4! 2! 3! 4! The first set of summands can be simplified while x2 x 3 x 4 the second set can be factored as follows: x 3 6 10 ... 1 2x 3 x2 4 x 3 5 x 4 ... 2! 3! 4! and simplifying the first set while factoring the second 2 2 3 4 x x x x set results to 1 2xx 1 ... . 2 2! 3! 4! 1x x234 x x ... 1 Since 1 x x23 x ... , for 11 x , 1 x x2 x 2 x 3 x 4 xx 1 ... . which when both sides are differentiated gives 2 2! 3! 4! 1 2 Hence, 2 1 2xx 3 ..., for , and since (1 x ) 2 1 x x xx23 E() x x e exx 1 ..., for all values of x (see [1] (1 x ) 2 2! 3! 2 and [2]), it follows that 12xx x E() x e 2 12 x 1 x x E( x )2 1 2 x e 23x (1 x ) 2 2 (2x x x ) e Ex() , 2 2(1 x ) 1 2 4xx x E() x2 e which completes the proof. (1 x ) 2 2 (1 x )22 (2 4 x x ) ex CONCLUSIONS Ex() 2 A factoriangular number is defined as the sum of 2(1 x ) corresponding factorial and triangular number and is 2 3 4 x 2 (2 5x 2 x x ) e given by nT! , where T n( n 1)/ 2 . Succession Ex() n n 2(1 x )2 of these numbers forms the sequence of factoriangular and the proof is completed. numbers: If the factoriangular number for n = 0 is included, 2, 5, 12, 34, 135, 741, 5068, 40356, 362925, ... . then the sequence of factoriangular numbers for n ≥ 0 is These factoriangular numbers follow the recurrence given by relations: {Ft } 1,2,5,12,34,135,741,5068,40356,362925,.... 2 n n 2 In this case, the sequence has the following EGF: Ftnn1 ( n 1) Ft , for n 1, and 2 Theorem 4. The exponential generating function of Ftn for n ≥ 0 is given by the closed formula nn2 21 23x Ft n Ft , for n 2 . 2 (2x x x ) e nn1 Ex() 2 2(1 x ) The exponential generating function of the sequence for 11 x . of factoriangular numbers is given by the formula: Proof: 2 (2 5x2 2 x 3 x 4 ) ex For the sequence 1, 2, 5, 12, 34, ..., the egf is Ex() 2(1 x )2 x2 x 3 x 4 E( x ) 1 2 x 5 12 34 .... for n 1 and . If the sequence of 2! 3! 4! factoriangular numbers is considered to begin with
168 P-ISSN 2350-7756 | E-ISSN 2350-8442 | www.apjmr.com Asia Pacific Journal of Multidisciplinary Research, Vol. 3, No. 4, November 2015 Castillo, Recurrence Relations and Generating Functions of the Sequence… ______n 0 , then the sequence becomes 1, 2, 5, 12, 34, 135, 741, 5068, 40356, 362925, ... and the exponential generating function is 2 (2x x23 x ) ex Ex() 2(1 x ) for n 0 and 11 x .
REFERENCES [1] Peterson, T. S. (1960). Calculus with Analytic Geometry. New York: Harper & Row, Publishers, Incorporated. [2] Leithold, L. (1996). The Claculus 7. Reading, Massachusetts: Addison Wesley Longman, Inc. [3] Sloane, N. J. A., Plouffe, S. (1995). The Encyclopedia of Integer Sequences. Academic Press. [4] Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences. http://oeis.org. [5] Castillo, R. C. (2015). On the sum of corresponding factorials and triangular numbers: Some preliminary results. Asia Pacific Journal of Multidisciplinary Research, 3(4.1), 5-11. [6] Castillo, R. C. (2015). On the sum of corresponding factorials and triangular numbers: Runsums, trapezoids and politeness. Asia Pacific Journal of Multidisciplinary Research, 3(4.2), 95-101. [7] Castillo, R. C. (2015). Sums of two triangulars and of two squares associated with sum of corresponding factorial and triangular number. Asia Pacific Journal of Multidisciplinary Research, 3(4.3), 28-36.
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