a(n,m) tabf head (staircase) for A128502 First convolution of Chebyshev S-polynomials sum(S(n-k,x)*S(k,x),k=0..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 2 0 0 0 0 0 0 0 2 3 -2 0 0 0 0 0 0 3 4 -6 0 0 0 0 0 0 4 5 -12 3 0 0 0 0 0 5 6 -20 12 0 0 0 0 0 6 7 -30 30 -4 0 0 0 0 7 8 -42 60 -20 0 0 0 0 8 9 -56 105 -60 5 0 0 0 9 10 -72 168 -140 30 0 0 0 10 11 -90 252 -280 105 -6 0 0 11 12 -110 360 -504 280 -42 0 0 12 13 -132 495 -840 630 -168 7 0 13 14 -156 660 -1320 1260 -504 56 0 14 15 -182 858 -1980 2310 -1260 252 -8 15 16 -210 1092 -2860 3960 -2772 840 -72 . . . G.f. for column m sequences: ((-1)^m)*(m+1)*(x^(2*m))/(1-x)^(m+2), m>=0. The column sequences, divided by (m+1)*(-1)^m coincide with the columns m+1 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)^2. The convolution polynomials S1(n,x):=sum(S(n-k,x)*S(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)^2. Row sums (signed) are: [1, 2, 1, -2, -4, -2, 3, 6, 3, -4, -8, -4, 5, 10, 5, -6,...] = A099254(n), n>=0. G.f.: 1/(1-x+x^2)^2 (convolution of the period 6 sequence A099254). Row sums (unsigned) are: [1, 2, 5, 10, 20, 38, 71, 130, 235, 420, 744, 1308, 2285, 3970, 6865, 11822,...] = A001629(n+2),n>=0. G.f.: 1/(1-x-x^2)^2 (Fibonacci convolution A001629). ####################################### e.o.f. ###########################################