[seqfan] Lack of takers: DoItYourself: the difference between A078521and A137432

[wouter meeussen]

and yet, the deeper into it, the "mysterouser" it gets:


w=24;
a= (CoefficientList[ #, r]&/@ CoefficientList[Series[Product[(1-z^k)^r, {k,
1, w}], {z, 0, w}], z])Range[0, w]!;
b=Table[ n! CoefficientList[SeriesCoefficient[ Series[1/(1 - t - t^2)^x, {t,
0, w}], n], x], {n, 0, w}];
ColumnForm[it=Table[Part[b,k]-Part[Abs/@a,k],{k,1+w}] ]
(Tr/@ it)/Range[0,w]!

u=GCD@@@ (DeleteCases[#,0]&/@ Drop[it,5])
{120, 720, 2520, 6720, 15120, 30240, 55440, 95040, 154440, 240240, 360360,
524160}

v=Range[5,16]!
{120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, \
87178291200, 1307674368000, 20922789888000}


v/u
{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800}

Range[0,11]!
{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800}

z=Take[#,{2,-5}]&/@ Drop[it,5];

quod=MapIndexed[#1 (-1+Tr[#2])! /((5-1+Tr[#2])!)&,z,1]

{1},
{1, 1},
{6, 5, 1},
{24, 35, 12, 1},
{168, 278, 131, 22, 1},
{1260, 2454, 1525, 365, 35, 1},
{12240, 25764, 18604, 5865, 835, 51, 1},
{123480, 299004, 249130, 94969, 17920, 1666, 70, 1},
{1572480, 3913776, 3615660, 1603868, 373009, 46424, 3010, 92, 1},
{21228480, 56405952, 57028932, 28560104, 7820253, 1212897, 106218, 5046,
117, 1},
{324172800, 903417120, 973641816, 538594100, 168425270, 31060785, 3416133,
220650, 7980, 145, 1},
{5428684800, 15670640160, 17966486736, 10784738316, 3762277420, 799192295,
105300888, 8591583, 424380, 12045, 176, 1}

Never seen this one before,
penultimate in each row are A000326  : Pentagonal numbers: n*(3*n-1)/2.

summed with alternating signs:
now it's getting definitely coockoo:
Map[Table[(-1)^k,{k,Length[#]}] . # &,quod]
{-1, 0, -2, 0, 0, 0, 0, 5040, 0, 0, 3628800, 0, 0, 0, 0, 0,
0, -355687428096000, 0, 0}


{0,2,7,10,17}!
{1, 2, 5040, 3628800, 355687428096000}

why 0,2,7,10,17 ? Looks like I'm LOST.
where is the DHARMA-initiative when you need it?

Wouter.



----- Original Message ----- 
From: "wouter meeussen" <wouter.meeussen at pandora.be>
To: "SeqFan" <seqfan at list.seqfan.eu>
Sent: Sunday, August 28, 2011 11:05 PM
Subject: [seqfan] help needed interpreting the difference between A078521and
A137432


>
> any takers?
>
> Wouter.
>