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begins:\n  0:        1;\n  1:        1,        1;\n  2:        3,        4,        1;\n  3:       15,       23,        9,        1;\n  4:      105,      176,       86,       16,       1;\n  5:      945,     1689,      950,      230,      25,      1;\n  6:    10395,    19524,    12139,     3480,     505,     36,     1;\n  7:   135135,   264207,   177331,    57379,   10045,    973,    49,    1;\n  8:  2027025,  4098240,  2924172,  1038016,  208054,  24640,  1708,   64,  1;\n  9: 34459425, 71697105, 53809164, 20570444, 4574934, 626934, 53676, 2796, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + x)*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] = (2*n - 1 + x)*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":"A028338","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:        3,        4,        1;\n  3:       15,       23,        9,        1;\n  4:      105,      176,       86,       16,       1;\n  5:      945,     1689,      950,      230,      25,      1;\n  6:    10395,    19524,    12139,     3480,     505,     36,     1;\n  7:   135135,   264207,   177331,    57379,   10045,    973,    49,    1;\n  8:  2027025,  4098240,  2924172,  1038016,  208054,  24640,  1708,   64,  1;\n  9: 34459425, 71697105, 53809164, 20570444, 4574934, 626934, 53676, 2796, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + x)*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + x)*v[i - 1]); 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":"A109692","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: 1,  4,    3;\n  3: 1,  9,   23,    15;\n  4: 1, 16,   86,   176,    105;\n  5: 1, 25,  230,   950,   1689,     945;\n  6: 1, 36,  505,  3480,  12139,   19524,    10395;\n  7: 1, 49,  973, 10045,  57379,  177331,   264207,   135135;\n  8: 1, 64, 1708, 24640, 208054, 1038016,  2924172,  4098240,  2027025;\n  9: 1, 81, 2796, 53676, 626934, 4574934, 20570444, 53809164, 71697105, 34459425;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2*n-1+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + x)*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + x)*v[i - 1]); 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-4f6e747b794bcc2e","steps":[],"edges":[],"text":"A028338"},"headline":"A028338: Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","chain":null}