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begins:\n  0: 1;\n  1: 0, 2;\n  2: 0, 1, 3;\n  3: 0, 0, 4,  4;\n  4: 0, 0, 1, 10,  5;\n  5: 0, 0, 0,  6, 20,  6;\n  6: 0, 0, 0,  1, 21, 35,   7;\n  7: 0, 0, 0,  0,  8, 56,  56,   8;\n  8: 0, 0, 0,  0,  1, 36, 126,  84,   9;\n  9: 0, 0, 0,  0,  0, 10, 120, 252, 120, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) + (x-x^2)*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] + (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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + (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":"A034867","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  2;\n  2:  3,   1;\n  3:  4,   4;\n  4:  5,  10,   1;\n  5:  6,  20,   6;\n  6:  7,  35,  21,   1;\n  7:  8,  56,  56,   8;\n  8:  9,  84, 126,  36,  1;\n  9: 10, 120, 252, 120, 10;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = 2*x*P(n-1,x) + (x-x^2)*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] + (x - x^2)*P[n - 2, x]];\nTable[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {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] + (x - x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":false}},{"oeis_id":"A109447","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2;\n   3:  1,   3;\n   4:  4,   4;\n   5:  1,  10,   5;\n   6:  6,  20,   6;\n   7:  1,  21,  35,   7;\n   8:  8,  56,  56,   8;\n   9:  1,  36, 126,  84,  9;\n  10: 10, 120, 252, 120, 10;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = 2*x*P(n-1,x) + (x-x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + (x - x^2)*P[n - 2, x]];\nTable[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = 2*x*v[i - 1] + (x - x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":false}},{"oeis_id":"A119900","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: 0, 1, 3;\n  3: 0, 0, 4,  4;\n  4: 0, 0, 1, 10,  5;\n  5: 0, 0, 0,  6, 20,  6;\n  6: 0, 0, 0,  1, 21, 35,   7;\n  7: 0, 0, 0,  0,  8, 56,  56,   8;\n  8: 0, 0, 0,  0,  1, 36, 126,  84,   9;\n  9: 0, 0, 0,  0,  0, 10, 120, 252, 120, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) + (x-x^2)*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] + (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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + (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":"A202064","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:  3,   1,   0;\n  3:  4,   4,   0,   0;\n  4:  5,  10,   1,   0,  0;\n  5:  6,  20,   6,   0,  0, 0;\n  6:  7,  35,  21,   1,  0, 0, 0;\n  7:  8,  56,  56,   8,  0, 0, 0, 0;\n  8:  9,  84, 126,  36,  1, 0, 0, 0, 0;\n  9: 10, 120, 252, 120, 10, 0, 0, 0, 0, 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) + (x-x^2)*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] + (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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + (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-e8e64448d4c6498a","steps":[],"edges":[],"text":"A119900"},"headline":"A119900: Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.","chain":null}