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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":"e7ebad11deb29a25861f191625caa54c315228059b02e3d7b15b0568f7f976ba","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f771467984d796d1","role":"sequence","name":"basis_map_A013609 of Exponential Riordan array (1, x(1+x/2)).","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:     1;\n  1:     3,      2;\n  2:     8,     14,       4;\n  3:    22,     66,      48,       8;\n  4:    66,    272,     348,     144,      16;\n  5:   206,   1100,    2020,    1480,     400,      32;\n  6:   688,   4452,   10680,   11560,    5520,    1056,     64;\n  7:  2388,  18284,   54264,   78120,   56000,   18816,   2688,    128;\n  8:  8732,  76736,  270928,  488768,  474320,  241024,  60032,   6656,   256;\n  9: 33076, 329112, 1348848, 2927904, 3605616, 2513952, 948864, 182016, 16128, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-4-2*x)*P(n-1,x) + (1+2*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n - 4 - 2*x)*P[n - 1, x] + (1 + 2*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(n - 4 - 2*x)*v[i - 1] + (1 + 2*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-f6268da8d9636534","steps":["basis A013609","D+1"],"edges":["e-8afcaf02015990bf","e-b08c78106b47f842"],"text":"A122848 \u2192 basis A013609 \u2192 D+1"},"headline":"A122848 \u2192 basis A013609 \u2192 D+1","chain":null}