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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":"8c61709833be0b4f6ce6eed6b8fe6353eba988567789eb591678f81122566132","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-c13c144ce0d4f57d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: 3-Stirling numbers of the second kind.","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:       0;\n   2:       1;\n   3:       9,        2;\n   4:      64,       30,        3;\n   5:     427,      314,       66,       4;\n   6:    2821,     2870,      945,     120,       5;\n   7:   18890,    24752,    11280,    2240,     195,      6;\n   8:  129683,   208852,   122808,   33740,    4585,    294,     7;\n   9:  918081,  1758094,  1274154,  452004,   84910,   8484,   420,   8;\n  10: 6721438, 14933034, 12912231, 5654544, 1371615, 189252, 14574, 576, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(19+8*x+x^2)*P(n-2,x) - (4+5*x+x^2)*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(19 + 8*x + x^2)*P[n - 2, x] - (4 + 5*x + x^2)*D[P[n - 2, x], x] + (9 + 2*x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(19 + 8*x + x^2)*v[i - 2] - (4 + 5*x + x^2)*deriv(v[i - 2], x) + (9 + 2*x)*v[i - 1] + (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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["x \u21a6 x+1","D"],"edges":["e-6768e07b6c7820a7","e-c1e588c2c6f796dc"],"text":"A143495 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A143495 \u2192 x \u21a6 x+1 \u2192 D","chain":null}