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begins:\n   1:   1;\n   2:   1,    1;\n   3:   2,    3,    1;\n   4:   4,    8,    5,    1;\n   5:   8,   20,   18,    7,    1;\n   6:  16,   48,   56,   32,    9,    1;\n   7:  32,  112,  160,  120,   50,   11,   1;\n   8:  64,  256,  432,  400,  220,   72,  13,   1;\n   9: 128,  576, 1120, 1232,  840,  364,  98,  15,  1;\n  10: 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(n-2)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 1])/(n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A081277","target":"reversed_triangle","first_row":1,"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: 1,  1;\n  2: 1,  3,   2;\n  3: 1,  5,   8,   4;\n  4: 1,  7,  18,  20,    8;\n  5: 1,  9,  32,  56,   48,   16;\n  6: 1, 11,  50, 120,  160,  112,   32;\n  7: 1, 13,  72, 220,  400,  432,  256,   64;\n  8: 1, 15,  98, 364,  840, 1232, 1120,  576,  128;\n  9: 1, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];\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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1])/(n - 1)); 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":1,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A200139","target":"triangle","first_row":1,"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:   1,    1;\n  2:   2,    3,    1;\n  3:   4,    8,    5,    1;\n  4:   8,   20,   18,    7,    1;\n  5:  16,   48,   56,   32,    9,    1;\n  6:  32,  112,  160,  120,   50,   11,   1;\n  7:  64,  256,  432,  400,  220,   72,  13,   1;\n  8: 128,  576, 1120, 1232,  840,  364,  98,  15,  1;\n  9: 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];\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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1])/(n - 1)); 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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-f2a2a3d7ea29bcbf","steps":[],"edges":[],"text":"A200139"},"headline":"A200139: Triangle T(n,k), read by rows, given 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