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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":"352646e934cb2a1db0f4b819a0ef77229ce2e78c1c2576572729501e0d801135","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-9c403911703f1b61","role":"sequence","name":"basis_map_A013610 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:     4,      3;\n   3:    15,     30,       9;\n   4:    46,    165,     144,      27;\n   5:   137,    696,    1098,     594,      81;\n   6:   392,   2583,    5976,    5832,    2268,     243;\n   7:  1095,   8862,   27099,   39150,   27297,    8262,     729;\n   8:  3002,  28809,  109656,  211383,  218538,  117855,   29160,    2187;\n   9:  8113,  90048,  410292,  991980, 1373922, 1097388,  481140,  100602,   6561;\n  10: 21676, 273147, 1449648, 4220856, 7362576, 7888266, 5111748, 1885194, 341172, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+6*x+9*x^2)*P(n-4,x) - (4+18*x+18*x^2)*P(n-3,x) - (2+6*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 4 + 3*x, P(3,x) = 15 + 30*x + 9*x^2, P(4,x) = 46 + 165*x + 144*x^2 + 27*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 3*x;\nP[3, x_] := 15 + 30*x + 9*x^2;\nP[4, x_] := 46 + 165*x + 144*x^2 + 27*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 9*x^2)*P[n - 4, x] - (4 + 18*x + 18*x^2)*P[n - 3, x] - (2 + 6*x + 9*x^2)*P[n - 2, x] + (4 + 6*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] = 4 + 3*x; v[3] = 15 + 30*x + 9*x^2; v[4] = 46 + 165*x + 144*x^2 + 27*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 6*x + 9*x^2)*v[i - 4] - (4 + 18*x + 18*x^2)*v[i - 3] - (2 + 6*x + 9*x^2)*v[i - 2] + (4 + 6*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 A013610","D+1"],"edges":["e-7f7d545e75881e09","e-5cb2c981549b3d71"],"text":"A122542 \u2192 basis A013610 \u2192 D+1"},"headline":"A122542 \u2192 basis A013610 \u2192 D+1","chain":null}