[seqfan] Re: DHARMA: the difference between A078521 and A137432

[Meeussen Wouter (bkarnd)]

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

just the pentagonal numbers - 5.
In[1]:= Table[n*(3*n - 1)/2, {n, -7, 7}] // Sort
Out[1]= {0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77}
In[2]:= Drop[ % - 5 , 3]
Out[2]={0, 2, 7, 10, 17, 21, 30, 35, 46, 52, 65, 72}

Remark that A078521 is just a signed version of A008298 (Triangle of D'Arcais numbers)
and that A137432 has been recycled since equivalent to A039692 (Jabotinsky-triangle).

Restating:

Is it self-evident that these two triangles
 A(w,r): Product(k=1..w; (1-z^k)^(-r) )
 B(w,r): (1-z-z^2)^(-r)
have a difference triangle that is so 'clean'?

{0}, 
{0, 0}, 
{0, 0, 0}, 
{0, 0, 0, 0}, 
{0, 0, 0, 0, 0}, 
{0, 120, 0, 0, 0, 0}, 
{0, 720, 720, 0, 0, 0, 0}, 
{0, 15120, 12600, 2520, 0, 0, 0, 0}, 
{0, 161280, 235200, 80640, 6720, 0, 0, 0, 0}

of which the 'core' 
(drop 5 rows, drop first and last 4 in each row)
simplifies to

{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}

no mysteries in the alternating sign row-sums,
if we (accept | prove) that this operation on A039692 (Jabotinsky-triangle)
produces -1,1,2,0,0,0,0...

Wouter.

-----------------------------------------------------
or in Mma speak:
w = 16;
a=(CoefficientList[#1, r] & ) /@ 
    CoefficientList[Series[Product[(1 - z^k)^-r, 
       {k, 1, w}], {z, 0, w}], z]*Range[0, w]! ;

b=(CoefficientList[#1, r] & ) /@ 
    CoefficientList[Series[ (1 -z-z^2)^-r, 
        {z, 0, w}], z]*Range[0, w]! ;

diff = Table[( b[[k]] - a[[k]]), {k, w}];
clip = Take[#, {2, -5}] & /@ Drop[diff, 5];
quod = MapIndexed[#1 (-1 + Tr[#2])!/((5 - 1 + Tr[#2])!) &, clip, 1]
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