%I A133156
%S A133156 1,2,4,1,8,4,16,12,1,32,32,6,64,80,24,1,128,192,80,8,256,448,
%T A133156 240,40,1,512,1024,672,160,10,1024,2304,1792,560,60,1,2048,5120,
%U A133156 4608,1792,280,12,4096,11264,11520,5376,1120,84,1
%V A133156 1,2,4,-1,8,-4,16,-12,1,32,-32,6,64,-80,24,-1,128,-192,80,-8,256,-448,
%W A133156 240,-40,1,512,-1024,672,-160,10,1024,-2304,1792,-560,60,-1,2048,-5120,
%X A133156 4608,-1792,280,-12,4096,-11264,11520,-5376,1120,-84,1
%N A133156 Irregular triangle read by rows: coefficients of U(n,x), Chebyshev polynomials 
               of the second kind with exponents in decreasing order.
%C A133156 The Chebyshev polynomials of the second kind are defined by the recurrence 
               relation: U(0,x) = 1; U(1,x) = 2x; U(n+1,x) = 2x*U(n,x) - U(n-1,x).
%C A133156 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 28 2008: 
               (Start)
%C A133156 Triangle read by rows, unsigned = A000012 * A028297
%C A133156 Row sums of absolute values give the Pell series, A000129. (End)
%C A133156 The row sums are: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,...}.
%D A133156 Wikipedia, (Chebyshev polynomials).
%D A133156 Tracale Austin, Hans Bantilan, Isao Jonas and Paul Kory, The Pfaffian 
               Transformation, Journal of Integer Sequences, Vol. 11 (2008), page 
               25 [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Dec 19 2008]
%F A133156 A generating function for U(n) is 1/(1 - 2tx + t^2). Given A038207, shift 
               down columns to allow for (1, 1, 2, 2, 3, 3,...) terms in each row, 
               then insert alternate signs.
%F A133156 t(n,m) = (-1)^m*Binomial[n - m, m]*2^(n - 2*m) [From Roger L. Bagula 
               and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 19 2008]
%e A133156 The first few Chebyshev polynomials of the second kind are:
%e A133156 1;
%e A133156 2x;
%e A133156 4x^2 - 1;
%e A133156 8x^3 - 4x;
%e A133156 16x^4 - 12x^2 + 1;
%e A133156 32x^5 - 32x^3 + 6x;
%e A133156 64x^6 - 80x^4 + 24x^2 - 1;
%e A133156 128x^7 - 192x^5 + 80x^3 - 8x;
%e A133156 256x^8 - 448x^6 + 240x^4 - 40x^2 + 1;
%e A133156 512x^9 - 1024x^7 _ 672x^5 - 160x^3 + 10x;
%e A133156 ...
%e A133156 Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Dec 19 2008: (Start)
%e A133156 {1},
%e A133156 {2},
%e A133156 {4, -1},
%e A133156 {8, -4},
%e A133156 {16, -12, 1},
%e A133156 {32, -32, 6},
%e A133156 {64, -80, 24, -1},
%e A133156 {128, -192, 80, -8},
%e A133156 {256, -448, 240, -40, 1},
%e A133156 {512, -1024, 672, -160,10},
%e A133156 {1024, -2304, 1792, -560, 60, -1} (End)
%t A133156 Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Dec 19 2008: (Start)
%t A133156 t[n_, m_] = (-1)^m*Binomial[n - m, m]*2^(n - 2*m);
%t A133156 Table[Table[t[n, m], {m, 0, Floor[n/2]}], {n, 0, 10}];
%t A133156 Flatten[%] (End)
%Y A133156 Cf. A038207, A053117.
%Y A133156 A018297, A000129 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 28 
               2008]
%Y A133156 Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, 
               [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 12 2009]
%Y A133156 Sequence in context: A121685 A125810 A152195 this_sequence A127529 A091977 
               A112829
%Y A133156 Adjacent sequences: A133153 A133154 A133155 this_sequence A133157 A133158 
               A133159
%K A133156 tabf,sign
%O A133156 0,2
%A A133156 Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 16 2007
%E A133156 More terms from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 12 2009