[seqfan] Re: A triangular observation (nothing proved)

[Richard Mathar]

The triangle constructed in http://list.seqfan.eu/pipermail/seqfan/2010-March/003828.html

   0,
   1,   1,
   2,   0,   4,
   3,   0,   0,   9,
   4,   2,   2,   2,  16,
   5,   2,   6,   6,   4,  25,
   6,   3,   3,  12,   9,   9,  36,
   7,   6,   8,  10,  22,  20,  18,  49,

has a 2nd column T(n,2) = 1,0,0,2,2,3,6,7,9,13,15,18,23,26,30,...
apparently following the rule a(n)= 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5),
g.f. -x*(1+x^3-2*x+x^2)/(1+x+x^2)/(x-1)^3.

It has a 3rd column T(n,3) =
4,0,2,6,3,8,13,12,21,28,30,44,54,60,80,94,105,132,151,168,203,228,252,296,328,
360,414,454,495
apparently following
a(n)= 2*a(n-1) -a(n-2) +2*a(n-3) -4*a(n-4) +2*a(n-5) -a(n-6) 2*a(n-7) -a(n-8)
g.f. -x^2*(-4-6*x^2+6*x^3-9*x^4+4*x^5+8*x)/(1+x+x^2)^2/(x-1)^4.

The generating functions indicate that these are higher order differences of
some linear combination of 1/(1+x+x^2), A049347, 1/(1+x+x^2)^2 variant
of A099470 and A099254, etc.
Likely, the other columns have homogeneous linear recurrences with constant
coefficients that increase their order by 3 each time a new column is
addressed.  This is a sort of punishment of using r=3 instead of the modest
Pascal triangle's r=1 :-)

The main diagonal is n^2 by definition.
The first sub-diagonal is T(n,n-1)=
1,0,0,2,4,9,18,29,45,67,93,126,167,214,270,336,410,495,592,699,819,953,1099,1260,1437,1628,1836,
which seems to obey
a(n)= 3*a(n-1) -3*a(n-2) +2*a(n-3) -3*a(n-4) +3*a(n-5) -a(n-6),
G.f. x*(1+x^4-3*x+3*x^2)/(1+x+x^2)/(x-1)^4

The second sub-diagonal is T(n,n-2)=
2,0,2,6,9,20,35,54,87,128,180,254,342,450,590,752,945,1182,1451,1764,2135,
2550,3024,3572,4178
where apparently
a(n)= 3*a(n-1) -3*a(n-2) +3*a(n-3) -6*a(n-4) +6*a(n-5) -3*a(n-6) +3*a(n-7) -3*a(n-8) +a(n-9).
associated with
g.f. -x*(2+8*x^2+9*x^4-7*x^5+2*x^6-6*x-6*x^3)/(1+x+x^2)^2/(x-1)^5 .

Again one could guess that the power of (1+x+x^2) and (x-1) in the
denominator of the g.f. increase by 1 each time one walks further
away from the diagonal.

The middle column is T(2n,n) = 
0,2,12,18,60,150,384,976,2268,6120,15020,34842,93636,229104,533076,1425762,3492228,
8150412,21740364,53332546,124803084,332408574,816661104,1915415952,5097429768,
12540083306,29468886084,78387875226,193067939780,454458266814
with no apparent recurrence with constant or polynomial (hypergeometric)
coefficients.

Richard J. Mathar

T := proc(n,k)
        if k <0 or k>n or n < 0 then
                0;
        elif k = 0 then
                n;
        elif k = n then
                n^2 ;
        else
                add( procname(n-3,j),j=k-3..k) ;
        end if;
end proc:
for n from 0 to 20 do
        for k from 0 to n do
                printf("%4d,",T(n,k)) ;
        end do ;
        printf("\n") ;
end do: