a(n,m)  tabf head (staircase) for  A128505
 
  Third convolution of Chebyshev S-polynomials

  sum(S(n-k,x)*S2(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m) ,m=0..floor(n/2)),
  with the second convolution sequence S2(k,x) with coefficient array 
  A128503.
  

  The row length sequence of this array is [1,1,2,2,3,3,4,4,...] = A004526.


    n\m      0        1        2         3        4        5      6 ...


    0        1        0        0         0        0        0      0

    1        4        0        0         0        0        0      0

    2       10       -4        0         0        0        0      0

    3       20      -20        0         0        0        0      0

    4       35      -60       10         0        0        0      0

    5       56     -140       60         0        0        0      0

    6       84     -280      210       -20        0        0      0

    7      120     -504      560      -140        0        0      0

    8      165     -840     1260      -560       35        0      0

    9      220    -1320     2520     -1680      280        0      0

   10      286    -1980     4620     -4200     1260      -56      0

   11      364    -2860     7920     -9240     4200     -504      0

   12      455    -4004    12870    -18480    11550    -2520     84

   13      560    -5460    20020    -34320    27720    -9240    840
    .
    .
    .



G.f. for column m sequences:  ((-1)^m)*binomial(m+3,3)*(x^(2*m))/(1-x)^(m+4), m>=0.

The column sequences, divided by binomial(m+3,3)*(-1)^m coincide with the columns 
m+3 of Pascal's triangle.


Row polynomials P3(n,x):= sum(a(n,m)*x^n,m=0..floor(n/2)) (increasing powers of x)
 are generated by 1/(1-z-x*z^2)^4.

The convolution polynomials S3(n,x):=sum(S(n-k,x)*S2(k,x),k=0..n)= 
sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)) are generated by  1/(1-x*z+z^2)^4.


Row sums (signed) are: 
[1, 4, 6, 0, -15, -24, -6, 36, 60, 20, -70, -120, -45,...]
= A128506(n),n>=0.

G.f.:  1/(1-x+x^2)^4 


Row sums (unsigned) are: 

[1, 4, 14, 40, 105, 256, 594, 1324, 2860, 6020, 12402, 25088, 49963, 98160,...]= A001872(n).  	

G.f.:  1/(1-x-x^2)^4 (Fibonacci convolution).  


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