bogus OGF in A002113 (and very many others)

Mon May 12 21:25:17 CEST 2008

sum(k=1,N, z^(2*k^2)*(1+z^k)/(1-z^k) )
sum((-1)^(k-1)*q^(k+1 choose 2)/(1-q^k), k, 1, inf)
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Subject: Re: bogus OGF in A002113 (and very many others)
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   simon.plouffe at sympatico.ca
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Subject: number of partitions bench mark
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Date: Mon, 12 May 2008 15:14:40 -0700
From: "Joshua Zucker" <joshua.zucker at gmail.com>
To: "Richard Mathar" <mathar at strw.leidenuniv.nl>
Subject: Re: number of partitions bench mark
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I didn't try Mathematica but it was the work of only a few minutes in
DrScheme to write the pentagonal number recurrence.  I'd share the
source code but I used an ugly hack - I precomputed the pentagonals up
to a large enough value and precomputed the signs to go with them, and
then just hacked away at the recursion, using memoization to avoid
recomputing old values in the recurrence.  So, ugly code.

I confirm PARI's answer:
11269 -> 231139177231303975514411787649455628959060199360109972557851519105155176180318215891795874905318274163248033071850
partitions

The other values are below in case you would like to check them with yours.

--Joshua Zucker

On Mon, May 12, 2008 at 1:48 PM, Richard Mathar
<mathar at strw.leidenuniv.nl> wrote:
> 11269
231139177231303975514411787649455628959060199360109972557851519105155176180318215891795874905318274163248033071850

> 11566
7958672699445426405771440953515770658326151510699702307294816461579682754354595845201915634351344747017524402065248

> 12376
99017248385851177305125749331254268692069636143943149618007509765629954605089512903256058142212835650280024690475548286

> 12430
183612308490711490511878455859461240643253822575808980147499525550386204601658427998766363882153765438515528391557517695

> 12700
3946960024003488206503243567381587243804394006082319578908117050087243536539117481873104881489885328405561972417839541991

> 12754
7261756919070420409788060626043794642848087861175771277772129563557760822282992173328641540319508310452599831276460527945

> 15013
300666632778197471994806746126441610313495695080917295105045094033586808265364200491184553315961495445684959489918201339252743943581

> 17589
45335612459012933951815830878480775442091783820061717187868448879353797428614153762560165042711595475166891868680787157479271019971074124934090

> 17797
332929387543783698188554831905365899632197136605414673273774323670408101035006940336102998080581855276321815894424234935283527166847977628681783

> 18181
12818757702041882484779137310430826655678863286672423907472403232930518897262027858430557766315506364507441468711804652014942054247921824233352554

> 18421
123122324382911993703590660204413462755544899702466104239165318515062106172029700112471908057442472133274039788590791987501667665046014995515542577

> 18453
166285217690859860353591934893763079216506301406346061243268616906698124878779091648526979885863786311572944145198678170336764360431521130012234959

> 18549
409006786049325622827405735992639595959766309097423230120155932606903214493209303558016570996557353883648380675787477004737872638563809056149522200

> 18597
640902278700248899624521933641642146919890082108532310522186090904028405189639086721065853619341219248416397768347773673511651859192559468349798015

> 18885
9374332371307145539758088248722898369219100068925600196668117726342341231963378315670499914153786963327212592119594171154767255117565909314829383335

> 18949
16969147251317539990826729770177839733020241962352760924387192459466175628601756014248547267216700799283065354778247643688233964117176155324099517500

> 18997
26464634284974117982023809131659117556789020579732726400422200079423004603720594044224174832086219572151505039334730543704825522863005736607113022912

And here's the ugly source code:
(require (planet "memoize.ss" ("dherman" "memoize.plt" 2 1)))

(define (generate-pentagonals k done)
  (cond
    [(>= k done) empty]
    [(= 1 (remainder k 3)) (generate-pentagonals (add1 k) done)]
    [else (cons (* 1/6 k (add1 k)) (generate-pentagonals (add1 k) done))]))

(define PENTAGONALS (generate-pentagonals 1 1000))

(define SIGNED-PENTAGONALS (build-list (length PENTAGONALS)
                                       (lambda (k) (list (expt -1
(floor (* k 1/2)))
                                                         (list-ref
PENTAGONALS k)))))

(define/memo (part-pentagonal n)
  (cond
    [(= n 0) 1]
    [else (apply + (map (lambda (p) (* (first p) (part-pentagonal (- n
(second p)))))
                        (filter (lambda (p) (<= (second p) n))
SIGNED-PENTAGONALS)))]))