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v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A090582","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:       2,        1;\n   3:       6,        6,        1;\n   4:      24,       36,       14,        1;\n   5:     120,      240,      150,       30,        1;\n   6:     720,     1800,     1560,      540,       62,       1;\n   7:    5040,    15120,    16800,     8400,     1806,     126,      1;\n   8:   40320,   141120,   191520,   126000,    40824,    5796,    254,     1;\n   9:  362880,  1451520,  2328480,  1905120,   834120,  186480,  18150,   510,    1;\n  10: 3628800, 16329600, 30240000, 29635200, 16435440, 5103000, 818520, 55980, 1022, 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 (1+2*x+x^2)*P(n,x) = (x+2*x^2+x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[Together[((x + 2*x^2 + x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\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] = ((x + 2*x^2 + x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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":"A131689","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,   2;\n  3: 0, 1,   6,     6;\n  4: 0, 1,  14,    36,     24;\n  5: 0, 1,  30,   150,    240,    120;\n  6: 0, 1,  62,   540,   1560,   1800,     720;\n  7: 0, 1, 126,  1806,   8400,  16800,   15120,    5040;\n  8: 0, 1, 254,  5796,  40824, 126000,  191520,  141120,   40320;\n  9: 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (x+2*x^2+x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[Together[((x + 2*x^2 + x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\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] = ((x + 2*x^2 + x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^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}}],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["reverse (shift +1)","x \u21a6 x/(x+1)"],"edges":["e-61bc78adfe1eb25a","e-e273b6c51200d112"],"text":"A173018 \u2192 reverse (shift +1) \u2192 x \u21a6 x/(x+1)"},"headline":"A173018 \u2192 reverse (shift +1) \u2192 x \u21a6 x/(x+1)","chain":null}