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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":"0e495cad24f4f3e44874fba25521d31f361dd7f3b96aa40058e64216ca575e14","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-84dfb3e427b94d98","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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:     1;\n   2:     3,      2;\n   3:    11,     16,      4;\n   4:    33,     82,     52,      8;\n   5:    97,    336,    368,    144,      16;\n   6:   275,   1226,   1936,   1312,     368,     32;\n   7:   763,   4160,   8612,   8496,    4112,    896,     64;\n   8:  2081,  13418,  34420,  44936,   31728,  11872,   2112,   128;\n   9:  5601,  41696, 127680, 208096,  195232, 106496,  32384,  4864,   256;\n  10: 14915, 125906, 448256, 877312, 1031776, 748864, 331392, 84736, 11008, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+4*x+4*x^2)*P(n-4,x) - (4+12*x+8*x^2)*P(n-3,x) - (2+4*x+4*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 11 + 16*x + 4*x^2, P(4,x) = 33 + 82*x + 52*x^2 + 8*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[3, x_] := 11 + 16*x + 4*x^2;\nP[4, x_] := 33 + 82*x + 52*x^2 + 8*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 4*x^2)*P[n - 4, x] - (4 + 12*x + 8*x^2)*P[n - 3, x] - (2 + 4*x + 4*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 11 + 16*x + 4*x^2; v[4] = 33 + 82*x + 52*x^2 + 8*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 4*x + 4*x^2)*v[i - 4] - (4 + 12*x + 8*x^2)*v[i - 3] - (2 + 4*x + 4*x^2)*v[i - 2] + (4 + 4*x)*v[i - 1]); 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-b0fc620949fb5990","steps":["basis A013609","D+1"],"edges":["e-778f192f6d6aff27","e-913556720ed92dd0"],"text":"A122542 \u2192 basis A013609 \u2192 D+1"},"headline":"A122542 \u2192 basis A013609 \u2192 D+1","chain":null}