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A112552 A modified Chebyshev transform of the second kind.

%I A112552 #16 Feb 07 2024 09:25:30
%S A112552 1,0,1,-2,0,1,0,-3,0,1,3,0,-4,0,1,0,6,0,-5,0,1,-4,0,10,0,-6,0,1,0,-10,
%T A112552 0,15,0,-7,0,1,5,0,-20,0,21,0,-8,0,1,0,15,0,-35,0,28,0,-9,0,1,-6,0,35,
%U A112552 0,-56,0,36,0,-10,0,1,0,-21,0,70,0,-84,0,45,0,-11,0,1,7,0,-56,0,126,0,-120,0,55,0,-12,0,1
%N A112552 A modified Chebyshev transform of the second kind.
%C A112552 Row sums are A112553.
%C A112552 Inverse is A112554.
%C A112552 Riordan array product (1/(1+x^2), x)*(1/(1+x^2), x/(1+x^2)).
%H A112552 G. C. Greubel, <a href="/A112552/b112552.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A112552 Riordan array (1/(1+x^2)^2, x/(1+x^2)).
%F A112552 T(n, k) = (-1)^floor((n-k)/2)*Sum_{j=0..n} (1+(-1)^(n-j))*(1+(-1)^(j-k))*binomial((j+k)/2, k)/4.
%F A112552 Unsigned triangle = A128174 * A149310, as infinite lower triangular matrices, with row sums A052952: (1, 1, 3, 4, 8, 12, 21, 33, ...). - _Gary W. Adamson_, Oct 28 2007
%F A112552 T(n, k) = (-1)^floor((n-k)/2)*((1 + (-1)^(n+k))/2)*binomial((n+k+2)/2, k+1). - _G. C. Greubel_, Jan 13 2022
%F A112552 T(n,k) = A049310(n+1,k+1) . - _R. J. Mathar_, Feb 07 2024
%e A112552 Triangle begins as:
%e A112552    1;
%e A112552    0,   1;
%e A112552   -2,   0,   1;
%e A112552    0,  -3,   0,   1;
%e A112552    3,   0,  -4,   0,   1;
%e A112552    0,   6,   0,  -5,   0,   1;
%e A112552   -4,   0,  10,   0,  -6,   0,  1;
%e A112552    0, -10,   0,  15,   0,  -7,  0,  1;
%e A112552    5,   0, -20,   0,  21,   0, -8,  0,   1;
%e A112552    0,  15,   0, -35,   0,  28,  0, -9,   0,   1;
%e A112552   -6,   0,  35,   0, -56,   0, 36,  0, -10,   0, 1;
%e A112552    0, -21,   0,  70,   0, -84,  0, 45,   0, -11, 0, 1;
%t A112552 Table[(-1)^Floor[(n-k)/2]*((1+(-1)^(n+k))/2)*Binomial[(n+k+2)/2, k+1], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 13 2022 *)
%o A112552 (Magma) [(-1)^Floor((n-k)/2)*((1+(-1)^(n+k))/2)*Binomial(Floor((n+k+2)/2), k+1): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jan 13 2022
%o A112552 (Sage) flatten([[(-1)^floor((n-k)/2)*((1+(-1)^(n+k))/2)*binomial((n+k+2)/2, k+1) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jan 13 2022
%Y A112552 Cf. A049310, A052952, A112553, A112554, A128174.
%K A112552 easy,sign,tabl
%O A112552 0,4
%A A112552 _Paul Barry_, Sep 13 2005