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begins:\n  0: 1;\n  1: 1,  3,   1;\n  2: 1,  7,  13,    7,     1;\n  3: 1, 11,  41,   63,    41,     11,      1;\n  4: 1, 15,  85,  231,   321,    231,     85,     15,       1;\n  5: 1, 19, 145,  575,  1289,   1683,   1289,    575,     145,      19,       1;\n  6: 1, 23, 221, 1159,  3649,   7183,   8989,   7183,    3649,    1159,     221,     23,      1;\n  7: 1, 27, 313, 2047,  8361,  22363,  40081,  48639,   40081,   22363,    8361,   2047,    313,     27,     1;\n  8: 1, 31, 421, 3303, 16641,  56695, 134245, 224143,  265729,  224143,  134245,  56695,  16641,   3303,   421,   31,   1;\n  9: 1, 35, 545, 4991, 29961, 124515, 369305, 795455, 1256465, 1462563, 1256465, 795455, 369305, 124515, 29961, 4991, 545, 35, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x+x^2)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x + x^2)*P[n - 1, 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 + 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + x^2)*v[i - 1] - 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":"A297191","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  3,   1;\n  2: 1,  7,  13,    7,     1;\n  3: 1, 11,  41,   63,    41,     11,      1;\n  4: 1, 15,  85,  231,   321,    231,     85,     15,       1;\n  5: 1, 19, 145,  575,  1289,   1683,   1289,    575,     145,      19,       1;\n  6: 1, 23, 221, 1159,  3649,   7183,   8989,   7183,    3649,    1159,     221,     23,      1;\n  7: 1, 27, 313, 2047,  8361,  22363,  40081,  48639,   40081,   22363,    8361,   2047,    313,     27,     1;\n  8: 1, 31, 421, 3303, 16641,  56695, 134245, 224143,  265729,  224143,  134245,  56695,  16641,   3303,   421,   31,   1;\n  9: 1, 35, 545, 4991, 29961, 124515, 369305, 795455, 1256465, 1462563, 1256465, 795455, 369305, 124515, 29961, 4991, 545, 35, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x+x^2)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x + x^2)*P[n - 1, 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 + 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + x^2)*v[i - 1] - 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":"A297191","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  3,   1;\n  2: 1,  7,  13,    7,     1;\n  3: 1, 11,  41,   63,    41,     11,      1;\n  4: 1, 15,  85,  231,   321,    231,     85,     15,       1;\n  5: 1, 19, 145,  575,  1289,   1683,   1289,    575,     145,      19,       1;\n  6: 1, 23, 221, 1159,  3649,   7183,   8989,   7183,    3649,    1159,     221,     23,      1;\n  7: 1, 27, 313, 2047,  8361,  22363,  40081,  48639,   40081,   22363,    8361,   2047,    313,     27,     1;\n  8: 1, 31, 421, 3303, 16641,  56695, 134245, 224143,  265729,  224143,  134245,  56695,  16641,   3303,   421,   31,   1;\n  9: 1, 35, 545, 4991, 29961, 124515, 369305, 795455, 1256465, 1462563, 1256465, 795455, 369305, 124515, 29961, 4991, 545, 35, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+4*x+x^2)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x + x^2)*P[n - 1, x] - x^2*P[n - 2, x]];\nTable[Reverse[CoefficientList[P[n, x], x]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + x^2)*v[i - 1] - x^2*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":false}}],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A008288"],"edges":["e-054f97e0823e3ab6"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008288"},"headline":"A297191: Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.","chain":"basis-map coefficients of A297191, shifted \u2192 basis A008288"}