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v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A054142","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  1;\n  2: 1,  3,   1;\n  3: 1,  5,   6,   1;\n  4: 1,  7,  15,  10,    1;\n  5: 1,  9,  28,  35,   15,    1;\n  6: 1, 11,  45,  84,   70,   21,   1;\n  7: 1, 13,  66, 165,  210,  126,  28,   1;\n  8: 1, 15,  91, 286,  495,  462, 210,  36,  1;\n  9: 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1;\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) = (1+2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = x + x^2.","mathematica":"P[0, x_] := x;\nP[1, x_] := x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] - x^2*P[n - 2, x]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] - x^2*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":true}},{"oeis_id":"A085478","target":"reversed_triangle","first_row":1,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  1;\n  2: 1,  3,   1;\n  3: 1,  6,   5,   1;\n  4: 1, 10,  15,   7,    1;\n  5: 1, 15,  35,  28,    9,    1;\n  6: 1, 21,  70,  84,   45,   11,   1;\n  7: 1, 28, 126, 210,  165,   66,  13,   1;\n  8: 1, 36, 210, 462,  495,  286,  91,  15,  1;\n  9: 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1;\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) = (1+2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = x + x^2.","mathematica":"P[0, x_] := x;\nP[1, x_] := x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] - x^2*P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] - x^2*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":true}},{"oeis_id":"A121314","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, 1;\n  2: 0, 1,  1;\n  3: 0, 1,  3,  1;\n  4: 0, 1,  5,  6,   1;\n  5: 0, 1,  7, 15,  10,   1;\n  6: 0, 1,  9, 28,  35,  15,   1;\n  7: 0, 1, 11, 45,  84,  70,  21,   1;\n  8: 0, 1, 13, 66, 165, 210, 126,  28,  1;\n  9: 0, 1, 15, 91, 286, 495, 462, 210, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, 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] = x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] - 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":"A165253","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,  0;\n  2: 1,  1,   0;\n  3: 1,  3,   1,   0;\n  4: 1,  6,   5,   1,   0;\n  5: 1, 10,  15,   7,   1,   0;\n  6: 1, 15,  35,  28,   9,   1,  0;\n  7: 1, 21,  70,  84,  45,  11,  1,  0;\n  8: 1, 28, 126, 210, 165,  66, 13,  1, 0;\n  9: 1, 36, 210, 462, 495, 286, 91, 15, 1, 0;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*x)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, 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] = x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] - 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-3af57aaf5c129e4b","steps":[],"edges":[],"text":"A121314"},"headline":"A121314: Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","chain":null}