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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":"dd26a3f4e6ccbc27f37b855e04cf8f2e808ea6de7a7cc4fbba7193c4d4fb6c8d","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-80f3c43c239c22ff","role":"sequence","name":"reversal 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:  3,    2;\n  2:  6,   15,     5;\n  3: 10,   49,    59,    13;\n  4: 15,  116,   264,   207,     34;\n  5: 21,  230,   804,  1173,    680,     89;\n  6: 28,  407,  1972,  4362,   4662,   2142,    233;\n  7: 36,  665,  4200, 12735,  20470,  17232,   6555,    610;\n  8: 45, 1024,  8092, 31674,  69295,  87240,  60528,  19641,  1597;\n  9: 55, 1506, 14460, 70206, 197017, 335900, 347092, 204719, 57913, 4181;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3+n^2-n-(10*n^3-19*n^2-5*n+6)*x-(20*n^2-32*n+11)*x^2-(10*n-5)*x^3)*P(n,x) = -(2*n^3+5*n^2+n-2-(11*n^3-27*n^2-26*n+14)*x-(27*n^2-56*n+10)*x^2+(25*n^3-25*n^2-26*n+13)*x^3+(50*n^2-50*n-10)*x^4+(25*n-25)*x^5)*P(n-1,x) - ((4*n^2-4)*x-(11*n^2+12*n+17)*x^2-(40*n^2+67*n+39)*x^3-(25*n^2+105*n+66)*x^4-(50*n+65)*x^5-25*x^6)*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[Together[(-(2*n^3 + 5*n^2 + n - 2 - (11*n^3 - 27*n^2 - 26*n + 14)*x - (27*n^2 - 56*n + 10)*x^2 + (25*n^3 - 25*n^2 - 26*n + 13)*x^3 + (50*n^2 - 50*n - 10)*x^4 + (25*n - 25)*x^5)*P[n - 1, x] - ((4*n^2 - 4)*x - (11*n^2 + 12*n + 17)*x^2 - (40*n^2 + 67*n + 39)*x^3 - (25*n^2 + 105*n + 66)*x^4 - (50*n + 65)*x^5 - 25*x^6)*D[P[n - 1, x], x])/(-2*n^3 - n^2 + n + (10*n^3 - 19*n^2 - 5*n + 6)*x + (20*n^2 - 32*n + 11)*x^2 + (10*n - 5)*x^3)]];\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] = (-(2*n^3 + 5*n^2 + n - 2 - (11*n^3 - 27*n^2 - 26*n + 14)*x - (27*n^2 - 56*n + 10)*x^2 + (25*n^3 - 25*n^2 - 26*n + 13)*x^3 + (50*n^2 - 50*n - 10)*x^4 + (25*n - 25)*x^5)*v[i - 1] - ((4*n^2 - 4)*x - (11*n^2 + 12*n + 17)*x^2 - (40*n^2 + 67*n + 39)*x^3 - (25*n^2 + 105*n + 66)*x^4 - (50*n + 65)*x^5 - 25*x^6)*deriv(v[i - 1], x))/(-2*n^3 - n^2 + n + (10*n^3 - 19*n^2 - 5*n + 6)*x + (20*n^2 - 32*n + 11)*x^2 + (10*n - 5)*x^3)); 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":["reverse","D+1"],"edges":["e-0c6cc2be731e428d","e-91156694e29d0ed9"],"text":"A060920 \u2192 reverse \u2192 D+1"},"headline":"A060920 \u2192 reverse \u2192 D+1","chain":null}