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A101124 Number triangle associated to Chebyshev polynomials of first kind.

%I A101124 #27 Dec 28 2018 10:26:29
%S A101124 1,0,1,-1,1,1,0,1,2,1,1,1,7,3,1,0,1,26,17,4,1,-1,1,97,99,31,5,1,0,1,
%T A101124 362,577,244,49,6,1,1,1,1351,3363,1921,485,71,7,1,0,1,5042,19601,
%U A101124 15124,4801,846,97,8,1,-1,1,18817,114243,119071,47525,10081,1351,127,9,1,0,1,70226,665857,937444,470449,120126,18817,2024,161
%N A101124 Number triangle associated to Chebyshev polynomials of first kind.
%H A101124 Seiichi Manyama, <a href="/A101124/b101124.txt">Antidiagonals n = 0..139, flattened</a>
%H A101124 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A101124 Number triangle S(n, k)=T(n-k, k), k<n, S(n, n)=1, 0 otherwise, where T(n, k)=(n/2)sum{j=0..floor(n/2), C(n-j, j)(-1)^j*(2k)^(n-2j)}.
%F A101124 Columns have g.f. x^k(1-kx)/(1-2kx+x^2).
%F A101124 Also, square array if(n=0, 1, T(n, k)) read by antidiagonals.
%e A101124 As a number triangle, rows begin:
%e A101124   {1},
%e A101124   {0,1},
%e A101124   {-1,1,1},
%e A101124   {0,1,2,1},
%e A101124   ...
%e A101124 As a square array, rows begin
%e A101124    1, 1,  1,   1,    1, ...
%e A101124    0, 1,  2,   3,    4, ...
%e A101124   -1, 1,  7,  17,   31, ...
%e A101124    0, 1, 26,  99,  244, ...
%e A101124    1, 1, 97, 577, 1921, ...
%t A101124 T[n_, k_] := SeriesCoefficient[x^k (1 - k x)/(1 - 2 k x + x^2), {x, 0, n}];
%t A101124 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 12 2017 *)
%Y A101124 Columns include A001075, A001541, A001091, A001079, A023038, A011943.
%Y A101124 Row sums are A101125.
%Y A101124 Diagonal sums are A101126.
%Y A101124 Main diagonal gives A115066.
%Y A101124 Mirror of A322836.
%Y A101124 Cf. A053120.
%K A101124 easy,sign,tabl
%O A101124 0,9
%A A101124 _Paul Barry_, Dec 02 2004