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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":"2b7dae0be160dfa0584b627334820e03deb32254edaff502bb2400c8cebe6690","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-eba9e40af29eafd3","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","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:   1;\n  2:   4,    3;\n  3:  10,    9;\n  4:  19,   28,    5;\n  5:  31,   80,   25;\n  6:  46,  195,   96,   7;\n  7:  64,  413,  322,  49;\n  8:  85,  784,  938, 232,  9;\n  9: 109, 1368, 2394, 912, 81;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1-3*x+3*x^2-x^3)*P(n-6,x) + (6-12*x+6*x^2)*P(n-5,x) - (15-18*x+3*x^2)*P(n-4,x) + (20-12*x)*P(n-3,x) - (15-3*x)*P(n-2,x) + 6*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 4 + 3*x, P(3,x) = 10 + 9*x, P(4,x) = 19 + 28*x + 5*x^2, P(5,x) = 31 + 80*x + 25*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 4 + 3*x;\nP[3, x_] := 10 + 9*x;\nP[4, x_] := 19 + 28*x + 5*x^2;\nP[5, x_] := 31 + 80*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[-(1 - 3*x + 3*x^2 - x^3)*P[n - 6, x] + (6 - 12*x + 6*x^2)*P[n - 5, x] - (15 - 18*x + 3*x^2)*P[n - 4, x] + (20 - 12*x)*P[n - 3, x] - (15 - 3*x)*P[n - 2, x] + 6*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 4 + 3*x; v[4] = 10 + 9*x; v[5] = 19 + 28*x + 5*x^2; v[6] = 31 + 80*x + 25*x^2; for(n = 6, N, my(i = n + 1); v[i] = -(1 - 3*x + 3*x^2 - x^3)*v[i - 6] + (6 - 12*x + 6*x^2)*v[i - 5] - (15 - 18*x + 3*x^2)*v[i - 4] + (20 - 12*x)*v[i - 3] - (15 - 3*x)*v[i - 2] + 6*v[i - 1]); 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-801895951aa20f29","steps":["Veronese 2","D+1"],"edges":["e-6ce79ceb855f1a6c","e-0b2004c9988421fa"],"text":"A103406 \u2192 Veronese 2 \u2192 D+1"},"headline":"A103406 \u2192 Veronese 2 \u2192 D+1","chain":null}