A112552 A modified Chebyshev transform of the second kind.
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, 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, 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
Offset: 0
Examples
Triangle begins as: 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, 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, 0, -56, 0, 36, 0, -10, 0, 1; 0, -21, 0, 70, 0, -84, 0, 45, 0, -11, 0, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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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
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Mathematica
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 *) -
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
Formula
Riordan array (1/(1+x^2)^2, x/(1+x^2)).
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.
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
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
T(n,k) = A049310(n+1,k+1) . - R. J. Mathar, Feb 07 2024
Comments