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begins:\n  0: 1;\n  1: 0, 1;\n  2: 0, 1,   1;\n  3: 0, 1,   4,     1;\n  4: 0, 1,  11,    11,     1;\n  5: 0, 1,  26,    66,    26,      1;\n  6: 0, 1,  57,   302,   302,     57,     1;\n  7: 0, 1, 120,  1191,  2416,   1191,   120,     1;\n  8: 0, 1, 247,  4293, 15619,  15619,  4293,   247,   1;\n  9: 0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = n*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[n*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], 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] = n*x*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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":"A008292","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    1;\n   3: 1,    4,     1;\n   4: 1,   11,    11,      1;\n   5: 1,   26,    66,     26,       1;\n   6: 1,   57,   302,    302,      57,       1;\n   7: 1,  120,  1191,   2416,    1191,     120,      1;\n   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;\n   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;\n  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 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) = n*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[n*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = n*x*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A008292","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    1;\n   3: 1,    4,     1;\n   4: 1,   11,    11,      1;\n   5: 1,   26,    66,     26,       1;\n   6: 1,   57,   302,    302,      57,       1;\n   7: 1,  120,  1191,   2416,    1191,     120,      1;\n   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;\n   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;\n  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 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) = n*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[n*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = n*x*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A123125","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,   4,     1;\n  4: 0, 1,  11,    11,     1;\n  5: 0, 1,  26,    66,    26,      1;\n  6: 0, 1,  57,   302,   302,     57,     1;\n  7: 0, 1, 120,  1191,  2416,   1191,   120,     1;\n  8: 0, 1, 247,  4293, 15619,  15619,  4293,   247,   1;\n  9: 0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = n*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[n*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], 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] = n*x*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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":"A173018","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,   4,     1,     0;\n  4: 1,  11,    11,     1,      0;\n  5: 1,  26,    66,    26,      1,     0;\n  6: 1,  57,   302,   302,     57,     1,     0;\n  7: 1, 120,  1191,  2416,   1191,   120,     1,   0;\n  8: 1, 247,  4293, 15619,  15619,  4293,   247,   1, 0;\n  9: 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 0;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = n*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[n*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], 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] = n*x*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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-efd1fdf07e5468d8","steps":["reverse (shift +1)"],"edges":["e-61bc78adfe1eb25a"],"text":"A173018 \u2192 reverse (shift +1)"},"headline":"A123125: Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.","chain":"A173018 \u2192 reverse (shift +1)"}