A128503 - OEIS

%I #7 Aug 29 2019 17:40:34

%S 1,3,6,-3,10,-12,15,-30,6,21,-60,30,28,-105,90,-10,36,-168,210,-60,45,

%T -252,420,-210,15,55,-360,756,-560,105,66,-495,1260,-1260,420,-21,78,

%U -660,1980,-2520,1260,-168,91,-858,2970,-4620,3150,-756,28,105,-1092,4290,-7920,6930,-2520,252,120,-1365

%N Array for second (k=2) convolution of Chebyshev's S(n,x)=U(n,x/2) polynomials.

%C 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)) with the first convolution S1(n,x) given by array A128502.

%C Row polynomials P2(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).

%H W. Lang, <a href="/A128503/a128503.txt">First 13 rows and more.</a>

%F a(n,m)= binomial(n-m+2,2)*binomial(n-m,m)*(-1)^m, m=0..floor(n/2), n>=0.

%F a(n,m)= binomial(m+2,2)*binomial(n-m+2,m+2)*(-1)^m, m=0..floor(n/2), n>=0.

%F G.f. for S2(n,x): 1/(1-x*z+z^2)^3.

%F G.f. for P2(n,x): 1/(1-z+x*z^2)^3

%e n=4: [15,-30,6] stands for the polynomial S2(4,x) = 15*x^4-30*x^2+6 = 2*(S(4,x)*S1(0,x)+S(3,x)*S1(1,x))+S(2,x)*S1(2,x).

%e n=4:[15,-30,6] stands also for the row polynomial P2(4,x) = 15-30*x+6*x^2.

%e [1];[3];[6,-3];[10,-12];[15,-30,6];[21,-60,30];[28,-105,90,-10];...

%Y Row sums (signed array) give A128504. Unsigned row sums are A001628.

%Y Cf. A128502 (k=1 convolution). A128505 (k=3 convolution).

%K sign,tabf,easy

%O 0,2

%A _Wolfdieter Lang_ Apr 04 2007