a(n,m)  tabf head (staircase) for  A128503

  Second convolution (k=2) of Chebyshev's S(n,x)=U(n,x/2) polynomials.

  sum(S(k1,x)*S(k2,x)*S(k3,x),k1+k2+k3=n)= sum(a(n,m)*x^(n-2*m), m=0..floor(n/2)).

  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       7 ...


    0        1        0       0         0        0         0       0       0

    1        3        0       0         0        0         0       0       0

    2        6       -3       0         0        0         0       0       0

    3       10      -12       0         0        0         0       0       0

    4       15      -30       6         0        0         0       0       0

    5       21      -60      30         0        0         0       0       0

    6       28     -105      90       -10        0         0       0       0

    7       36     -168     210       -60        0         0       0       0

    8       45     -252     420      -210       15         0       0       0

    9       55     -360     756      -560      105         0       0       0

   10       66     -495    1260     -1260      420       -21       0       0

   11       78     -660    1980     -2520     1260      -168       0       0

   12       91     -858    2970     -4620     3150      -756      28       0

   13      105    -1092    4290     -7920     6930     -2520     252       0

   14      120    -1365    6006    -12870    13860     -6930    1260     -36

   15      136    -1680    8190    -20020    25740    -16632    4620    -360
    .
    .
    .


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

The column sequences, divided by binomial(m+2,2)*(-1)^m coincide with the columns m+2 of 
Pascal's triangle.


Row polynomials P(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)^3.

The convolution polynomials S2(n,x):=sum(S(n-k,x)*S1(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)^3.
Here S1(n,x) is the first (k=1) convolution of the S-polynomials with themselves, see 
coefficient array A128502.


Row sums (signed) are: [1, 3, 3, -2, -9, -9, 3, 18, 18, -4, -30, -30, 5, 45, 45, -6,...] = A128504(n), n>=0.

G.f.: 1/(1-x+x^2)^3 (second convolution of the period 6 sequence A099254).


Row sums (unsigned) are:[1, 3, 9, 22, 51, 111, 233, 474, 942, 1836, 3522, 6666, 12473, 23109, 42447, 77378,...] = A001628(n),n>=0.

G.f.:  1/(1-x-x^2)^3 (second Fibonacci convolution A001629).  


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