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%I #7 Nov 11 2017 11:52:01
%S 1,0,2,-2,0,3,0,-6,0,4,3,0,-12,0,5,0,12,0,-20,0,6,-4,0,30,0,-30,0,7,0,
%T -20,0,60,0,-42,0,8,5,0,-60,0,105,0,-56,0,9,0,30,0,-140,0,168,0,-72,0,
%U 10,-6,0,105,0,-280,0,252,0,-90,0,11,0,-42,0,280,0,-504,0,360,0,-110,0,12,7,0,-168,0,630,0,-840,0,495,0,-132,0,13
%N Convolution triangle for Chebyshev S polynomials (rising powers).
%C See the array A128502 without zeros and falling powers. This is the main entry.
%C The coefficient triangle for Chebyshev S polynomials is given in A049310.
%C The self-convolution (or first convolution) of the S polynomials is S1(n, x) := Sum_{k=0..n} S(k, x)*S(n-k, x), n >= 0, and S1(n, x) = Sum_{m=0..n} T(n, m)*x^m.
%F T(n, m) = [x^m] S1(n, x), with the first convolution S1 of the Chebyshev S polynomials. See a comment above.
%F T(n, m) = 0 if n-m is odd and T(n, m) = (-1)^((n-m)/2)*((n-m)/2 + 1)*binomial(n - (n-m)/2 +1, (n-m)/2 +1) = (-1)^((n-m)/2)*(n - (n-m)/2 + 1)* binomial(n - (n-m)/2, (n-m)/2) if n-m is even.
%F O.g.f. of {S1(n, x)}_{n >= 0} is G1(z,x) = (1/(1 - x*z + z^2))^2.
%e The triangle T(n, m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e 0: 1
%e 1: 0 2
%e 2: -2 0 3
%e 3: 0 -6 0 4
%e 4: 3 0 -12 0 5
%e 5: 0 12 0 -20 0 6
%e 6: -4 0 30 0 -30 0 7
%e 7: 0 -20 0 60 0 -42 0 8
%e 8: 5 0 -60 0 105 0 -56 0 9
%e 9: 0 30 0 -140 0 168 0 -72 0 10
%e 10: -6 0 105 0 -280 0 252 0 -90 0 11
%e 11: 0 -42 0 280 0 -504 0 360 0 -110 0 12
%e 12: 7 0 -168 0 630 0 -840 0 495 0 -132 0 13
%e ...
%Y Cf. A049310, A128502.
%K sign,tabl,easy
%O 0,3
%A _Wolfdieter Lang_, Nov 07 2017
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