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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":"7c10eb32bd7a655153b3d6a783f297578a6490dca4d5dadbf4942810639266aa","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:     0;\n   2:     2;\n   3:     8,      8;\n   4:    26,     56,     24;\n   5:    80,    256,    240,     64;\n   6:   234,    992,   1440,    832,    160;\n   7:   664,   3496,   6864,   6208,   2560,    384;\n   8:  1842,  11576,  28632,  35392,  22880,   7296,    896;\n   9:  5024,  36672, 109344, 171648, 152320,  76032,  19712,  2048;\n  10: 13522, 112384, 392064, 746624, 845120, 579840, 234752, 51200, 4608;\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) = 0, P(2,x) = 2, P(3,x) = 8 + 8*x, P(4,x) = 26 + 56*x + 24*x^2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[3, x_] := 8 + 8*x;\nP[4, x_] := 26 + 56*x + 24*x^2;\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] = 0; v[2] = 2; v[3] = 8 + 8*x; v[4] = 26 + 56*x + 24*x^2; 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"],"edges":["e-778f192f6d6aff27","e-44df082fff3d8adb"],"text":"A122542 \u2192 basis A013609 \u2192 D"},"headline":"A122542 \u2192 basis A013609 \u2192 D","chain":null}