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v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A053117","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  0,  2;\n  2: -1,  0,   4;\n  3:  0, -4,   0,    8;\n  4:  1,  0, -12,    0,  16;\n  5:  0,  6,   0,  -32,   0,   32;\n  6: -1,  0,  24,    0, -80,    0,   64;\n  7:  0, -8,   0,   80,   0, -192,    0,   128;\n  8:  1,  0, -40,    0, 240,    0, -448,     0, 256;\n  9:  0, 10,   0, -160,   0,  672,    0, -1024,   0, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] - v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A053118","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2, 0;\n  2:   4, 0,    -1;\n  3:   8, 0,    -4, 0;\n  4:  16, 0,   -12, 0,   1;\n  5:  32, 0,   -32, 0,   6, 0;\n  6:  64, 0,   -80, 0,  24, 0,   -1;\n  7: 128, 0,  -192, 0,  80, 0,   -8, 0;\n  8: 256, 0,  -448, 0, 240, 0,  -40, 0,  1;\n  9: 512, 0, -1024, 0, 672, 0, -160, 0, 10, 0;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = 2*x*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] - v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-ef9f4d5289b8feff","steps":[],"edges":[],"text":"chebyshev_u"},"headline":"chebyshev_u","chain":null}