number of partitions bench mark

Tue May 13 11:58:57 CEST 2008

> From: "Richard Mathar" <mathar at strw.leidenuniv.nl>
> Subject: number of partitions bench mark

> It might be helpful if someone with Mma
> could produce another suite of benchmarks for these n below as some
> kind of "third party" counter check:

For what it is worth, GBnums give the following results :

A070177 benchmark

10
42
100
190569292
1000
24061467864032622473692149727991
10000
36167251325636293988820471890953695495016030339315650422081868605887952568754066420592310556052906916435144

Your benchmark :

11269 (matches with PARI result)
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

Regards,
JT

---------------------------------------------
http://www.echolalie.com/gbnums
--------------------------------------------

----__JWM__J7d87.7295S.069fM

Seqfans,
Sum_{n>=3D0} log(1 + q^n*x)^n*y^n/n! =3D Sum_{n>=3D0} binomial(q^n*y,n)*x^n.
=20
I seek help in generalizing this result in the following way.=20
=20
=20
Find a triangle of integers {T(n,k), n>=3Dk>=3D1} such that:=20
(1) Sum_{n>=3D0} (1/n!)*Product_{k=3D1..n} log(1 + T(n,k)*x)
is a series in x with only integer coefficients;
=20
(2) T(n,k+1) > T(n,k) > 0 for k=3D1..n, n>=3D1.
=20
=20
Could someone find the triangle with the least positive T(n,k)=20
that satisifies these conditions?=20

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----__JWM__J7d87.7295S.069fM

<html><P>Seqfans,<BR>    Recall the identity that I shared i=
n a prior email:<BR>Sum_{n>=3D0} log(1 + q^n*x)^n*y^n/n! =3D Sum_{n>=
=3D0} binomial(q^n*y,n)*x^n.<BR> <BR>I seek help in generalizing this =
result in the following way. <BR> <BR> <BR>Find a triangle of int=
egers {T(n,k), n>=3Dk>=3D1} such that: </P>
<P>(1) Sum_{n>=3D0} (1/n!)*Product_{k=3D1..n} log(1 + T(n,k)*x)<BR>is a =
T(n,k) > 0 for k=3D1..n, n>=3D1.<BR> <BR> <BR>Could someone=
<font face=3DTimes-New-Roman size=3D2><br><br>_____________________________=
________________________________<br><a href=3D"http://thirdpartyoffers.juno=
.com/TGL2122/fc/Ioyw6i3oHgMrxh9Enoom0hb5LC0dXr1otkdBjDhHbPkdr2oj8QiORP/?cou=
nt=3D1234567890" target=3D_blank>Click here for great computer networking s=
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----__JWM__J7d87.7295S.069fM--

These errors in Maple seem bad enough to be placed on record,
next update) to preserve them.

Neil

> A082745=A064955

Yes, you are right and I will merge them.