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A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)).

%I A128505 #24 Sep 04 2019 02:06:53
%S A128505 1,4,10,-4,20,-20,35,-60,10,56,-140,60,84,-280,210,-20,120,-504,560,
%T A128505 -140,165,-840,1260,-560,35,220,-1320,2520,-1680,280,286,-1980,4620,
%U A128505 -4200,1260,-56,364,-2860,7920,-9240,4200,-504,455,-4004,12870,-18480,11550,-2520,84,560,-5460,20020,-34320
%N A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)).
%C A128505 S3(n,x) := Sum_{k=0..n} S(n-k,x)*S2(k,x) = Sum_{m=0..floor(n/2)} a(n,m)*x^(n-2*m)  with the second convolution S2(n,x) given by array A128503.
%C A128505 Row polynomials P3(n,x) :=  Sum_{m=0..floor(n/2)} a(n,m)*x^m (increasing powers of x).
%H A128505 Wolfdieter Lang, <a href="/A128505/a128505.txt">First 15 rows and more</a>.
%F A128505 a(n,m) = binomial(n-m+3,3)*binomial(n-m,m)*(-1)^m, m = 0..floor(n/2), n >= 0.
%F A128505 a(n,m) = binomial(m+3,3)*binomial(n-m+3,m+3)*(-1)^m, m = 0..floor(n/2), n >= 0.
%F A128505 G.f. for S3(n,x): 1/(1-x*z+z^2)^4.
%F A128505 G.f. for P3(n,x): 1/(1-z+x*z^2)^4.
%e A128505   1;
%e A128505   4;
%e A128505   10,   -4;
%e A128505   20,  -20;
%e A128505   35,  -60,  10;
%e A128505   56, -140,  60;
%e A128505   84, -280, 210,  -20;
%e A128505   120,-504, 560, -140;
%e A128505   ...
%e A128505 n=4: [35,-60,10] stands also for the row polynomial P3(4,x) = 35-60*x+10*x^2.
%Y A128505 Row sums (signed array) give A128506. Unsigned row sums are A001872.
%Y A128505 Cf. A128503 (k=2 convolution).
%K A128505 sign,tabf,easy
%O A128505 0,2
%A A128505 _Wolfdieter Lang_, Apr 04 2007
%E A128505 Name edited by _Petros Hadjicostas_, Sep 04 2019