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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":"c24b058f3cd9f467923a24d77fd6431a13ff2879c871003de0d62fd51c0096f6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9358d848e03a2727","role":"sequence","name":"derivative of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   0;\n  1:   0;\n  2:   2;\n  3:  12,    6;\n  4:  30,   60,    12;\n  5:  56,  210,   180,    20;\n  6:  90,  504,   840,   420,    30;\n  7: 132,  990,  2520,  2520,   840,    42;\n  8: 182, 1716,  5940,  9240,  6300,  1512,   56;\n  9: 240, 2730, 12012, 25740, 27720, 13860, 2520, 72;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-9*n+10)*P(n,x) = (2*n^2-5*n+3+(6*n^2-21*n+12)*x)*P(n-1,x) - ((n-1)*x+(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 + (6*n^2 - 21*n + 12)*x)*P[n - 1, x] - ((n - 1)*x + (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 10)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 3 + (6*n^2 - 21*n + 12)*x)*v[i - 1] - ((n - 1)*x + (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 10)); 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}}],"derivation":{"root":"n-6758fe7729845be8","steps":["D","D"],"edges":["e-23ce7e4146b3d83c","e-12890356e7b15b07"],"text":"A054142 \u2192 D \u2192 D"},"headline":"A054142 \u2192 D \u2192 D","chain":null}