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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":"95998f1ac278cde615419c14ee58e61293225385b92a6f9961fe6abf746273d0","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-d7230c9910e765ce","role":"sequence","name":"basis_map_A013611 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:     5,      4;\n   3:    19,     48,      16;\n   4:    59,    276,     304,       64;\n   5:   177,   1184,    2432,     1664,      256;\n   6:   509,   4436,   13504,    17152,     8448,     1024;\n   7:  1427,  15312,   61904,   117632,   106752,    40960,     4096;\n   8:  3923,  49988,  252208,   642624,   873728,   613376,   192512,    16384;\n   9: 10625, 156736,  948096,  3037952,  5560832,  5840896,  3334144,   884736,   65536;\n  10: 28437, 476580, 3361280, 12991488, 30030336, 42518528, 36233216, 17399808, 3997696, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+8*x+16*x^2)*P(n-4,x) - (4+24*x+32*x^2)*P(n-3,x) - (2+8*x+16*x^2)*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 5 + 4*x, P(3,x) = 19 + 48*x + 16*x^2, P(4,x) = 59 + 276*x + 304*x^2 + 64*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 4*x;\nP[3, x_] := 19 + 48*x + 16*x^2;\nP[4, x_] := 59 + 276*x + 304*x^2 + 64*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 8*x + 16*x^2)*P[n - 4, x] - (4 + 24*x + 32*x^2)*P[n - 3, x] - (2 + 8*x + 16*x^2)*P[n - 2, x] + (4 + 8*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] = 5 + 4*x; v[3] = 19 + 48*x + 16*x^2; v[4] = 59 + 276*x + 304*x^2 + 64*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 8*x + 16*x^2)*v[i - 4] - (4 + 24*x + 32*x^2)*v[i - 3] - (2 + 8*x + 16*x^2)*v[i - 2] + (4 + 8*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 A013611","D+1"],"edges":["e-458ebe1b0b79e88d","e-ca00db08e06ee680"],"text":"A122542 \u2192 basis A013611 \u2192 D+1"},"headline":"A122542 \u2192 basis A013611 \u2192 D+1","chain":null}