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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"b58b0e44b1baec33335e04290d90c4238852d8841863d87610ee9d9fdeced21e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7f247edfefc78f8c","role":"sequence","name":"basis_map_A013610 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    0;\n   2:    3;\n   3:    9,    18;\n   4:   21,    90,     81;\n   5:   45,   288,    567,    324;\n   6:   90,   774,   2349,   2916,    1215;\n   7:  174,  1872,   7695,  14904,   13365,    4374;\n   8:  327,  4230,  21951,  57996,   81405,   56862,   15309;\n   9:  603,  9108,  57105, 191484,  368145,  402408,  229635,  52488;\n  10: 1095, 18918, 139077, 565704, 1385100, 2077650, 1852389, 892296, 177147;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2-12*n+12-(9*n-18)*x)*P(n,x) = -(6*n^2-51*n+90-(45*n-144)*x+54*x^2)*P(n-1,x) + (5*n-15+(18*n-69)*x+(9*n-81)*x^2-27*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 - 51*n + 90 - (45*n - 144)*x + 54*x^2)*P[n - 1, x] + (5*n - 15 + (18*n - 69)*x + (9*n - 81)*x^2 - 27*x^3)*D[P[n - 1, x], x])/(3*n^2 - 12*n + 12 - (9*n - 18)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(6*n^2 - 51*n + 90 - (45*n - 144)*x + 54*x^2)*v[i - 1] + (5*n - 15 + (18*n - 69)*x + (9*n - 81)*x^2 - 27*x^3)*deriv(v[i - 1], x))/(3*n^2 - 12*n + 12 - (9*n - 18)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":"none","regular_triangle":false}}],"derivation":{"root":"n-392f49b496f79fbf","steps":["basis A013610","D"],"edges":["e-74fcd2d5aef5a8cd","e-eb2781729cf2bae0"],"text":"A026729 \u2192 basis A013610 \u2192 D"},"headline":"A026729 \u2192 basis A013610 \u2192 D","chain":null}