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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":"df9222a502dad632e01153ef24ddb035ec0f835bebd7d555c42197f2387ad91a","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-b3afa198ee370ac3","role":"sequence","name":"theta_plus_one of Riordan array (1, 2+2x).","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: 4;\n  2: 4, 24;\n  3: 0, 48,  96;\n  4: 0, 24, 288,  320;\n  5: 0,  0, 288, 1280,  960;\n  6: 0,  0,  96, 1920, 4800,  2688;\n  7: 0,  0,   0, 1280, 9600, 16128,   7168;\n  8: 0,  0,   0,  320, 9600, 40320,  50176,  18432;\n  9: 0,  0,   0,    0, 4800, 53760, 150528, 147456, 46080;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-(4*n-4)*x)*P(n,x) = -(2*n^2-8*n+6-(16*n-24)*x+24*x^2)*P(n-1,x) + ((4*n-4)*x+(2*n-18)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 8*n + 6 - (16*n - 24)*x + 24*x^2)*P[n - 1, x] + ((4*n - 4)*x + (2*n - 18)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(n^2 - n - (4*n - 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 8*n + 6 - (16*n - 24)*x + 24*x^2)*v[i - 1] + ((4*n - 4)*x + (2*n - 18)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(n^2 - n - (4*n - 4)*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}}],"derivation":{"root":"n-f5a001953ee28e8f","steps":["\u03b8+1","D"],"edges":["e-66f8fe9d623a70f6","e-e0bcbe94cb6b62f0"],"text":"A099040 \u2192 \u03b8+1 \u2192 D"},"headline":"A099040 \u2192 \u03b8+1 \u2192 D","chain":null}