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). ####################################### e.o.f. ###########################################