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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":"2b53b269fd78fd5aef71645d40cac53df41e06889473b3b01860ee94b782ebd7","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-75de16b1bf66665d","role":"sequence","name":"veronese_section_2_0 of Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","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:     2;\n  2:     6,      1;\n  3:    22,      9;\n  4:    85,     53,     1;\n  5:   322,    273,    20;\n  6:  1175,   1322,   193,    1;\n  7:  4132,   6112,  1400,   35;\n  8: 14070,  27043,  8755,  494,  1;\n  9: 46628, 114691, 49856, 4794, 54;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7-9*x-3*x^2+5*x^3)*P(n-6,x) - (4-12*x+12*x^2-4*x^3)*d/dx P(n-6,x) + (31-22*x+11*x^2-4*x^3)*P(n-5,x) + (28-44*x+20*x^2-4*x^3)*d/dx P(n-5,x) - (122-85*x+19*x^2)*P(n-4,x) - (68-84*x+16*x^2)*d/dx P(n-4,x) + (143-78*x+8*x^2)*P(n-3,x) + (68-76*x+8*x^2)*d/dx P(n-3,x) - (74-31*x)*P(n-2,x) - (28-28*x)*d/dx P(n-2,x) + (15-4*x)*P(n-1,x) + (4-4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 6 + x, P(3,x) = 22 + 9*x, P(4,x) = 85 + 53*x + x^2, P(5,x) = 322 + 273*x + 20*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 6 + x;\nP[3, x_] := 22 + 9*x;\nP[4, x_] := 85 + 53*x + x^2;\nP[5, x_] := 322 + 273*x + 20*x^2;\nP[n_, x_] := P[n, x] = Expand[(7 - 9*x - 3*x^2 + 5*x^3)*P[n - 6, x] - (4 - 12*x + 12*x^2 - 4*x^3)*D[P[n - 6, x], x] + (31 - 22*x + 11*x^2 - 4*x^3)*P[n - 5, x] + (28 - 44*x + 20*x^2 - 4*x^3)*D[P[n - 5, x], x] - (122 - 85*x + 19*x^2)*P[n - 4, x] - (68 - 84*x + 16*x^2)*D[P[n - 4, x], x] + (143 - 78*x + 8*x^2)*P[n - 3, x] + (68 - 76*x + 8*x^2)*D[P[n - 3, x], x] - (74 - 31*x)*P[n - 2, x] - (28 - 28*x)*D[P[n - 2, x], x] + (15 - 4*x)*P[n - 1, x] + (4 - 4*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; v[2] = 2; v[3] = 6 + x; v[4] = 22 + 9*x; v[5] = 85 + 53*x + x^2; v[6] = 322 + 273*x + 20*x^2; for(n = 6, N, my(i = n + 1); v[i] = (7 - 9*x - 3*x^2 + 5*x^3)*v[i - 6] - (4 - 12*x + 12*x^2 - 4*x^3)*deriv(v[i - 6], x) + (31 - 22*x + 11*x^2 - 4*x^3)*v[i - 5] + (28 - 44*x + 20*x^2 - 4*x^3)*deriv(v[i - 5], x) - (122 - 85*x + 19*x^2)*v[i - 4] - (68 - 84*x + 16*x^2)*deriv(v[i - 4], x) + (143 - 78*x + 8*x^2)*v[i - 3] + (68 - 76*x + 8*x^2)*deriv(v[i - 3], x) - (74 - 31*x)*v[i - 2] - (28 - 28*x)*deriv(v[i - 2], x) + (15 - 4*x)*v[i - 1] + (4 - 4*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-fac4d4a000d744e8","steps":["Veronese 2","D+1"],"edges":["e-a9acea761c71ee36","e-4829f4850d04edf5"],"text":"A060920 \u2192 Veronese 2 \u2192 D+1"},"headline":"A060920 \u2192 Veronese 2 \u2192 D+1","chain":null}