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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":"69ae5bbca54262450ab883ed59a5b013f39b51a890e617beb866e894817e47bf","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-7eb19ab4ebf03277","role":"sequence","name":"shift_x_plus_one of Stirling2 triangle with scaled diagonals (powers of 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:        0;\n  1:        1;\n  2:        8,        2;\n  3:       55,       30,        3;\n  4:      384,      332,       72,        4;\n  5:     2841,     3380,     1170,      140,       5;\n  6:    22504,    34022,    16440,     3140,     240,      6;\n  7:   190863,   348922,   217833,    58380,    7105,    378,     7;\n  8:  1728064,  3695512,  2834160,   989464,  169120,  14280,   560,   8;\n  9: 16640785, 40661352, 36987492, 16015944, 3578190, 423864, 26292, 792, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(13+6*x+x^2)*P(n-2,x) - (6+8*x+2*x^2)*d/dx P(n-2,x) + (8+2*x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(13 + 6*x + x^2)*P[n - 2, x] - (6 + 8*x + 2*x^2)*D[P[n - 2, x], x] + (8 + 2*x)*P[n - 1, x] + (2 + 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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(13 + 6*x + x^2)*v[i - 2] - (6 + 8*x + 2*x^2)*deriv(v[i - 2], x) + (8 + 2*x)*v[i - 1] + (2 + 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-a7e9ebb9860a9dbe","steps":["x \u21a6 x+1","D"],"edges":["e-6dbd3471b7b80662","e-38ff48d7eaa598a1"],"text":"A075497 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A075497 \u2192 x \u21a6 x+1 \u2192 D","chain":null}