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 A136523 Triangle T(n,k) = A053120(n,k)+A053120(n-1,k). 0
 1, 1, 1, -1, 1, 2, -1, -3, 2, 4, 1, -3, -8, 4, 8, 1, 5, -8, -20, 8, 16, -1, 5, 18, -20, -48, 16, 32, -1, -7, 18, 56, -48, -112, 32, 64, 1, -7, -32, 56, 160, -112, -256, 64, 128, 1, 9, -32, -120, 160, 432, -256, -576, 128, 256, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row sums are A040000(n). Gram-Schmidt vector analysis indicates this is orthogonal. Integration of products of the associated polynomials p_n(x) = sum_{k>=0} T(n,k)*x^k with the Chebyshev weight function 1/sqrt(1-x^2) over the interval (-1..1) gives it is tridiagonal orthogonal: Table[Table[Integrate[Sqrt[1/(1 - x^2)]*Q[x,n]*Q[x, m], {x, -1, 1}], {n, 0, 10}], {m, 0, 10}]; LINKS EXAMPLE 1; 1, 1; -1, 1, 2; -1, -3, 2, 4; 1, -3, -8, 4, 8; 1, 5, -8, -20, 8, 16; -1, 5, 18, -20, -48, 16, 32; -1, -7, 18, 56, -48, -112,32, 64; 1, -7, -32, 56,160, -112, -256, 64, 128; 1, 9, -32, -120, 160, 432, -256, -576, 128,256; -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512; MATHEMATICA Clear[B, x, n] (* A053120*) B[x, -1] = 0; B[x, 0] = 1; B[x, 1] = x; B[x_, n_] := B[x, n] = 2*x*B[x, n - 1] - B[x, n - 2]; Table[ExpandAll[B[x, n] + B[x, n - 1]], {n, 0, 10}]; a0 = Table[CoefficientList[B[x, n] + B[x, n - 1], x], {n, 0, 10}]; Flatten[a0] (* alternative definition*) Q[x, 0] = 1; Q[x, 1] = x + 1; Q[x_, n_] := Q[x, n] = B[x, n] + B[x, n - 1]; CROSSREFS Cf. A053120, A081277, A124182. Sequence in context: A007337 A167430 A056892 * A228731 A163507 A003963 Adjacent sequences:  A136520 A136521 A136522 * A136524 A136525 A136526 KEYWORD easy,tabl,sign AUTHOR Roger L. Bagula, Mar 23 2008 STATUS approved

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