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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":"f51954f2888533e527c6750f2027aa414dee1d78492abf5156ee5f070c529a8c","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-e7968148eadb4da9","role":"sequence","name":"basis_map_A007318 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:    1;\n  2:    2;\n  3:    9,    2;\n  4:   22,    6;\n  5:   68,   30,   3;\n  6:  174,   92,  12;\n  7:  475,  318,  66,   4;\n  8: 1214,  942, 240,  20;\n  9: 3147, 2846, 915, 120, 5;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (7*n-12+(2*n-4)*x)*P(n-1,x) - (39+25*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n - 12 + (2*n - 4)*x)*P[n - 1, x] - (39 + 25*x + 4*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 12 + (2*n - 4)*x)*v[i - 1] - (39 + 25*x + 4*x^2)*deriv(v[i - 1], x))/(n - 1)); 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-ac04e52510a48bb5","steps":["basis A007318","D"],"edges":["e-16890f4567488209","e-e43a4ecad5cd6484"],"text":"A249139 \u2192 basis A007318 \u2192 D"},"headline":"A249139 \u2192 basis A007318 \u2192 D","chain":null}