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begins:\n  0: 1;\n  1: 1,  2,   1;\n  2: 1,  4,   6,   4,    1;\n  3: 1,  6,  15,  20,   15,    6,     1;\n  4: 1,  8,  28,  56,   70,   56,    28,     8,     1;\n  5: 1, 10,  45, 120,  210,  252,   210,   120,    45,    10,     1;\n  6: 1, 12,  66, 220,  495,  792,   924,   792,   495,   220,    66,    12,     1;\n  7: 1, 14,  91, 364, 1001, 2002,  3003,  3432,  3003,  2002,  1001,   364,    91,   14,    1;\n  8: 1, 16, 120, 560, 1820, 4368,  8008, 11440, 12870, 11440,  8008,  4368,  1820,  560,  120,  16,   1;\n  9: 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 1]); 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":"A034870","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,  2,   1;\n  2: 1,  4,   6,   4,    1;\n  3: 1,  6,  15,  20,   15,    6,     1;\n  4: 1,  8,  28,  56,   70,   56,    28,     8,     1;\n  5: 1, 10,  45, 120,  210,  252,   210,   120,    45,    10,     1;\n  6: 1, 12,  66, 220,  495,  792,   924,   792,   495,   220,    66,    12,     1;\n  7: 1, 14,  91, 364, 1001, 2002,  3003,  3432,  3003,  2002,  1001,   364,    91,   14,    1;\n  8: 1, 16, 120, 560, 1820, 4368,  8008, 11440, 12870, 11440,  8008,  4368,  1820,  560,  120,  16,   1;\n  9: 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 1]); 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":"A034870","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,  2,   1;\n  2: 1,  4,   6,   4,    1;\n  3: 1,  6,  15,  20,   15,    6,     1;\n  4: 1,  8,  28,  56,   70,   56,    28,     8,     1;\n  5: 1, 10,  45, 120,  210,  252,   210,   120,    45,    10,     1;\n  6: 1, 12,  66, 220,  495,  792,   924,   792,   495,   220,    66,    12,     1;\n  7: 1, 14,  91, 364, 1001, 2002,  3003,  3432,  3003,  2002,  1001,   364,    91,   14,    1;\n  8: 1, 16, 120, 560, 1820, 4368,  8008, 11440, 12870, 11440,  8008,  4368,  1820,  560,  120,  16,   1;\n  9: 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 1, x]];\nTable[Reverse[CoefficientList[P[n, x], x]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 2*x + x^2)*v[i - 1]); 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-08f49539099949d5","steps":[],"edges":[],"text":"A034870"},"headline":"A034870: Even-numbered rows of Pascal's triangle.","chain":null}