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an exact degree step keeps every row nonzero; S_n = A + B D + C D^2 with A = 1 + x, B = 2 + 4 x, C = 4 x: Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. 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= 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (2 + 4*x)*deriv(v[i - 1], x) + 4*x*deriv(deriv(v[i - 1], x), 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":"A119743","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1,   1;\n   2: 1,   6,     3;\n   3: 1,  15,    45,     15;\n   4: 1,  28,   210,    420,      105;\n   5: 1,  45,   630,   3150,     4725,       945;\n   6: 1,  66,  1485,  13860,    51975,     62370,      10395;\n   7: 1,  91,  3003,  45045,   315315,    945945,     945945,     135135;\n   8: 1, 120,  5460, 120120,  1351350,   7567560,   18918900,   16216200,    2027025;\n   9: 1, 153,  9180, 278460,  4594590,  41351310,  192972780,  413513100,  310134825,   34459425;\n  10: 1, 190, 14535, 581400, 13226850, 174594420, 1309458150, 5237832600, 9820936125, 6547290750, 654729075;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + (2+4*x)*d/dx P(n-1,x) + 4*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1 + x.","mathematica":"P[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (2 + 4*x)*D[P[n - 1, x], x] + 4*x*D[P[n - 1, x], {x, 2}]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + (2 + 4*x)*deriv(v[i - 1], x) + 4*x*deriv(deriv(v[i - 1], x), 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, 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