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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":"72395cc437516b2f9185b9a1763d64b004887b4e4de87f88e6f6f7dfc15d6417","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9a6b0e2632270c41","role":"sequence","name":"basis_map_A008277 of Riordan array (1-x,x/(1-x)).","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:     3,       2;\n  1:    10,      14,        3;\n  2:    32,      76,       36,        4;\n  3:   100,     368,      294,       72,        5;\n  4:   308,    1672,     2022,      824,      125,       6;\n  5:   940,    7304,    12618,     7640,     1905,     198,       7;\n  6:  2852,   31096,    74046,    62664,    22885,    3870,     294,      8;\n  7:  8620,  130088,   416874,   474712,   238525,   58422,    7161,    416,     9;\n  8: 25988,  537592,  2279502,  3404104,  2263065,  753606,  132608,  12344,   567,  10;\n  9: 78220, 2202344, 12203898, 23463960, 20096585, 8744526, 2072679, 274992, 20124, 750, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(6+4*x+x^2)*P(n-2,x) - (2*x+x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 10 + 14*x + 3*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 10 + 14*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(6 + 4*x + x^2)*P[n - 2, x] - (2*x + x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; v[2] = 10 + 14*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(6 + 4*x + x^2)*v[i - 2] - (2*x + x^2)*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-7717563017cb9e5f","steps":["basis A008277","D"],"edges":["e-94a89e959b0863ff","e-0e677ea4848fcdb0"],"text":"A159854 \u2192 basis A008277 \u2192 D"},"headline":"A159854 \u2192 basis A008277 \u2192 D","chain":null}