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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":"12f888e4a7edf526a77bea61a9fc70b3f2f25775ca0abb41bea22f8490f704e0","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-6a690be6f27c7004","role":"sequence","name":"veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.","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: 1;\n  1: 1;\n  2: 1,  4,   3;\n  3: 3,  6,   3;\n  4: 1,  8,  18,  16,   5;\n  5: 5, 20,  30,  20,   5;\n  6: 1, 12,  45,  80,  75,  36,   7;\n  7: 7, 42, 105, 140, 105,  42,   7;\n  8: 1, 16,  84, 224, 350, 336, 196, 64, 9;\n  9: 9, 72, 252, 504, 630, 504, 252, 72, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n)*P(n,x) = -(n^3-3*n^2+2-(3*n^2-n-2)*x-(n^3-n)*x^2)*P(n-1,x) + (n^2-n-1+(n^2-3*n-2)*x-(n^2+3*n+1)*x^2-(n^2+n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 + 2 - (3*n^2 - n - 2)*x - (n^3 - n)*x^2)*P[n - 1, x] + (n^2 - n - 1 + (n^2 - 3*n - 2)*x - (n^2 + 3*n + 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 1, x], x])/(n^2 - n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 3*n^2 + 2 - (3*n^2 - n - 2)*x - (n^3 - n)*x^2)*v[i - 1] + (n^2 - n - 1 + (n^2 - 3*n - 2)*x - (n^2 + 3*n + 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 1], x))/(n^2 - n)); 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-6a690be6f27c7004","steps":["D"],"edges":["e-8e7e8819edb1ea3e"],"text":"Veronese parent of A109446 \u2192 D"},"headline":"Veronese parent of A109446 \u2192 D","chain":null}