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begins:\n  0:  1;\n  1:  1,   1;\n  2:  2,   4,    2;\n  3:  3,   9,    9,    3;\n  4:  5,  20,   30,   20,    5;\n  5:  8,  40,   80,   80,   40,    8;\n  6: 13,  78,  195,  260,  195,   78,   13;\n  7: 21, 147,  441,  735,  735,  441,  147,   21;\n  8: 34, 272,  952, 1904, 2380, 1904,  952,  272,  34;\n  9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+2*x+x^2)*P(n-2,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[(1 + x)*P[n - 1, x] + (1 + 2*x + x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {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] = (1 + x)*v[i - 1] + (1 + 2*x + x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":false}},{"oeis_id":"A016095","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:  1,   1;\n  2:  2,   4,    2;\n  3:  3,   9,    9,    3;\n  4:  5,  20,   30,   20,    5;\n  5:  8,  40,   80,   80,   40,    8;\n  6: 13,  78,  195,  260,  195,   78,   13;\n  7: 21, 147,  441,  735,  735,  441,  147,   21;\n  8: 34, 272,  952, 1904, 2380, 1904,  952,  272,  34;\n  9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+2*x+x^2)*P(n-2,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[(1 + x)*P[n - 1, x] + (1 + 2*x + x^2)*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (1 + 2*x + x^2)*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":"A016095","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:  1,   1;\n  2:  2,   4,    2;\n  3:  3,   9,    9,    3;\n  4:  5,  20,   30,   20,    5;\n  5:  8,  40,   80,   80,   40,    8;\n  6: 13,  78,  195,  260,  195,   78,   13;\n  7: 21, 147,  441,  735,  735,  441,  147,   21;\n  8: 34, 272,  952, 1904, 2380, 1904,  952,  272,  34;\n  9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + (1+2*x+x^2)*P(n-2,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[(1 + x)*P[n - 1, x] + (1 + 2*x + x^2)*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (1 + 2*x + x^2)*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-ed81f100961f7473","steps":[],"edges":[],"text":"basis-map coefficients of 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