Based on a previous technique deployed in some speciﬁc low order cases, we develop an automated computational procedure to evaluate instances within a class of inﬁnite series comprising exponentiated multiples of generalized linear combinations of Catalan numbers. The methodology is explained, and new results given.N/

Larcombe, Peter J.

O'Neill, Sam T.

Based on a previous technique deployed in some speciﬁc low order cases, we develop an automated computational procedure to evaluate instances within a class of inﬁnite series comprising exponentiated multiples of generalized linear combinations of Catalan numbers. The methodology is explained, and new results given

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A GENERATING FUNCTION APPROACH TO THE AUTOMATEDEVALUATION OF SUMS OF EXPONENTIATED MULTIPLES OFGENERALIZED CATALAN NUMBER LINEAR COMBINATIONSPETER J. LARCOMBE AND SAM T. O’NEILLAbstract. Based on a previous technique deployed in some specific low order cases, wedevelop an automated computational procedure to evaluate instances within a class of infi-nite series comprising exponentiated multiples of generalized linear combinations of Catalannumbers. The methodology is explained, and new results given.1. Introduction1.1. Background. Let cn =1n+1(2nn)denote the (n + 1)th term (n ≥ 0) of the sequence{c0, c1, c2, c3, c4, . . .} = {1, 1, 2, 5, 14, . . .} of Catalan numbers whose (ordinary) generatingfunction isG(x) =12x(1−√1− 4x) =∑n≥0cnxn. (1.1)This paper examines the evaluation of a class of seriesIp(β) = (−1)p∑n≥1βnhp(cn−1, . . . , cn+p−2)=∑n≥1βn[b(p)0 cn−1 + b(p)1 cn + b(p)2 cn+1 + · · ·+ b(p)p−1cn+p−2], (1.2)the summand of which comprises, for integer p ≥ 1, a particular p-term linear combination ofCatalan numbers with an exponentiated multiplier β 6= 0. Values of the constants b(p)0 , . . . , b(p)p−1are, from [2, Lemma 2, p. 216], known to have closed formb(p)r = (−1)r16p−(r+1)(2p− (r + 1)r), r = 0, . . . , p− 1, (1.3)being those (up to a sign) of the linear function hp(cn−1, . . . , cn+p−2) which, for any fixed p,arises in a power series expansion of the trigonometric function sin(2pα) according tosin(2pα)/2 =p∑n=1α(p)n sin2n−1(α) +∞∑n=1hp(cn−1, . . . , cn+p−2)22(n+p)−3sin2(n+p)−1(α) (1.4)in odd powers of sin(α).This type of power series was first discussed in 2000 by Larcombe [1] (where historicalcontext was provided), and more recently aspects of convergence have been examined [6] withthe natural principal range of convergence extended from |α| < π2 (seen in [1]) to |α| ≤π2analytically. Noting the parity of the sine function, it is found that the α interval (0, π2 ] in(1.4) translates to a range β ∈ (0, 14 ] in (1.2), and on this basis evaluations have been made forthe series S(β) = I1(β) in [3], and for T (β) =12I2(β) and U(β) = I3(β) in [4], using a variety ofmethods (that is, via direct evaluation of (1.4) itself, by appeal to hypergeometric theory, andMAY 2018 121THE FIBONACCI QUARTERLYthrough a generating function route that we extend here). A purely hypergeometric approachto tackle the completely general series Ip(β) for the first time can be seen in [5], where newevaluations of I4(β) and I5(β) have been given.We now outline the remit of the paper, before moving on to further sections that detail ourmethodology and results.1.2. This Paper. It is appropriate to list the first few series described by (1.2), which are,using (1.3),I1(β) =∑n≥1βn[cn−1],I2(β) = 2∑n≥1βn[8cn−1 − cn],I3(β) =∑n≥1βn[256cn−1 − 64cn + 3cn+1],I4(β) = 4∑n≥1βn[1024cn−1 − 384cn + 40cn+1 − cn+2],I5(β) =∑n≥1βn[65536cn−1 − 32768cn + 5376cn+1 − 320cn+2 + 5cn+3],I6(β) = 2∑n≥1βn[524288cn−1 − 327680cn + 73728cn+1 − 7168cn+2 + 280cn+3 − 3cn+4],I7(β) =∑n≥1βn[16777216cn−1 − 12582912cn + 3604480cn+1 − 491520cn+2+ 32256cn+3 − 896cn+4 + 7cn+5],I8(β) = 8∑n≥1βn[33554432cn−1 − 29360128cn + 10223616cn+1 − 1802240cn+2+ 168960cn+3 − 8064cn+4 + 168cn+5 − cn+6], (1.5)and so on. We develop one of the methods deployed in [4], which enables evaluations ofthose members of the series class (1.2) to be made easily in an automated fashion. While thehypergeometric formulation of [5] relies, using specialized algebraic software, on the symboliccalculation of a requisite (degree p) polynomial through which to deliver evaluations in anyp instance, ours here—based on the Catalan sequence generating function G(x) alone—is amuch simpler one to design theoretically and execute computationally; we are thus able torealize, in a straightforward manner, evaluations of series not considered to date.2. MethodologyFor any p ≥ 1 the series Ip(β) can be represented as a function of β, G(β) and the explicit p−1 Catalan numbers c0, . . . , cp−2. This is evident from manipulation of the final summand termof (1.2) into an appropriate form for use. Ignoring the accompanying constant for convenience,122 VOLUME 56, NUMBER 2EVALUATION OF SUMS OF EXPONENTIATED MULTIPLES OF CATALAN NUMBERSwe write ∑n≥1βncn+p−2 = β−(p−2)∞∑n=p−1βncn= β−(p−2)( ∞∑n=0βncn −p−2∑n=0βncn)= β−(p−2)(G(β)−p−2∑n=0βncn), (2.1)giving, for particular values p = 2, 3, 4, 5, . . . ,∑n≥1βncn = G(β)− 1,∑n≥1βncn+1 = β−1[G(β)− (1 + β)],∑n≥1βncn+2 = β−2[G(β)− (1 + β + 2β2)],∑n≥1βncn+3 = β−3[G(β)− (1 + β + 2β2 + 5β3)], (2.2)whose continuing pattern is clear. Expressing Ip(β) (1.2) concisely asIp(β) =p∑s=1∑n≥1βnb(p)s−1cn+s−2 , (2.3)then since (2.1) holds for p = 1 also (where, taking the sum∑−1n=0 βncn to be zero, it correctlyreads∑n≥1 βncn−1 = β−(−1)[G(β)− 0] = βG(β)), we arrive from (2.3) at the following aftera little work:Theorem 2.1. For p ≥ 1,Ip(β) =(p−1∑i=0b(p)i β1−i)G(β)−p−1∑i=1b(p)i β1−i i−1∑j=0cjβj .3. Results3.1. Closed Forms for Ip(β). The formula that is Theorem 2.1 is easily coded up, yieldingclosed forms (factored where required)I1(β) = βG(β),I2(β) = 2[(8β − 1)G(β) + 1],I3(β) = (256β − 64 + 3β−1)G(β) + 61− 3β−1,I4(β) = 4[(1024β − 384 + 40β−1 − β−2)G(β) + 346− 39β−1 + β−2],I5(β) = (65536β − 32768 + 5376β−1 − 320β−2 + 5β−3)G(β)+ 28007− 5066β−1 + 315β−2 − 5β−3,I6(β) = 2[(524288β − 327680 + 73728β−1 − 7168β−2 + 280β−3 − 3β−4)G(β)+ 266930− 67105β−1 + 6894β−2 − 277β−3 + 3β−4], (3.1)MAY 2018 123THE FIBONACCI QUARTERLYin the first six instances, though further ones are readily generated; this puts us in a positionto give some new results.3.2. New Results. In [5, Eqs. (1.8), (1.9), p. 261] evaluations were listed for Ip(β) (p = 1, 2, 3)at β = 14 ,316 ,18 and116 (interest in these particular values is explained in Section 1.2 therein),and further ones generated correspondingly for I4(β) and I5(β) (see (4.4),(4.5) of Section 4,p. 268). All of these are a reproducible consequence of (3.1) above (see [3,4] for its lower orderapplications in more detail), from which we can add additional ones as follows. In line withthe earlier work described, we find thatI6(1/4) = 186780,I6(3/16) = 131076,I6(1/8) = 83540,I6(1/16) = 40260, (3.2)andI7(1/4) = 2723234,I7(3/16) = 1412842090/729,I7(1/8) = 2(575237 + 32768√2),I7(1/16) = 2(3933277− 2097152√3),I8(1/4) = 40311776,I8(3/16) = 63323144416/2187,I8(1/8) = 18663968,I8(1/16) = 32(2097152√3− 3349019), (3.3)all having been tested numerically—to a low tolerance—against those convergent values of theseries of (1.5) that they describe in error-free form.4. SummarySome fairly routine analysis facilitates the automated evaluation of specific instances withina class of infinite series comprising exponentiated multiples of generalized Catalan numberlinear combinations, with some hitherto unseen results presented. The evaluating formulaestablished (Theorem 2.1)—being easy to derive and straightforward to code up—means thatthe methodology has a clear advantage over the hypergeometric approach taken previously [5]in terms of its relative level of complexity.This short article, together with [3–5], completes work undertaken on the evaluation of someunusual series that arise naturally within a topic whose historic origins are of mathematicalinterest [1,2] and have been recognized in a genealogy of the Catalan sequence (see the definitiveaccount by Igor Pak in Richard Stanley’s latest text [7, Appendix B, pp. 177–189]). It maybe the case that certain linear combinations (with or without an exponentiated multiplier) ofintegers drawn from sequences other than the Catalan sequence are equally summable—eitherby an existing technique such as that described here, or else through a new one—which is aquestion posed and left open for the attention of any reader.124 VOLUME 56, NUMBER 2EVALUATION OF SUMS OF EXPONENTIATED MULTIPLES OF CATALAN NUMBERSReferences[1] P. J. Larcombe, On Catalan numbers and expanding the sine function, Bulletin of the I.C.A., 28 (2000),39–47.[2] P. J. Larcombe, On a new formulation of Xinrong for the embedding of Catalan numbers in series formsof the sine function, J. Combinatorial Mathematics and Combinatorial Computing, 42 (2002), 209–221.[3] P. J. Larcombe, Closed form evaluations of some series involving Catalan numbers, Bulletin of the I.C.A.,71 (2014), 117–119.[4] P. J. Larcombe, Closed form evaluations of some series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations, The Fibonacci Quarterly, 53.3 (2015), 253–260.[5] P. J. Larcombe, On the evaluation of sums of exponentiated multiples of generalized Catalan number linearcombinations using a hypergeometric approach, The Fibonacci Quarterly, 54.3 (2016), 259–270.[6] P. J. Larcombe, S. T. O’Neill, and E. J. Fennessey, On certain series expansions of the sine function:Catalan numbers and convergence, The Fibonacci Quarterly, 52.3 (2014), 236–242.[7] R. P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015.MSC2010: 11K31, 05A15.Department of Computing and Mathematics, College of Engineering and Technology, Uni-versity of Derby, Kedleston Road, Derby DE22 1GB, England, U.K.E-mail address: p.j.larcombe@derby.ac.ukInstitute for Learning Enhancement and Innovation, University of Derby, Kedleston Road,Derby DE22 1GB, England, U.K.E-mail address: S.Oneill@derby.ac.ukMAY 2018 125

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A generating function approach to the automated evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations.

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A generating function approach to the automated evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations.

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