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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":"9df5299515543d676d82ee40a03f88e6cebea69580106c43aafc2b46878d0906","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-8652466ad45b52e6","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.","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:        3,        1;\n  3:       14,        9,        1;\n  4:       86,       87,       21,        1;\n  5:      657,      923,      369,       45,       1;\n  6:     5995,    10854,     6287,     1340,      93,      1;\n  7:    63541,   141354,   108945,    34900,    4500,    189,     1;\n  8:   766701,  2027629,  1966179,   862485,  172960,  14469,   381,   1;\n  9: 10372418, 31831473, 37359445, 21081569, 5941410, 799897, 45347, 765, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2+2*x)*P(n,x) = -(n^3-5*n^2+8*n-4+(2*n^3-9*n^2+9*n)*x+(n^3-n^2-9*n+11)*x^2+(3*n^2-11*n+12)*x^3+(2*n-4)*x^4)*P(n-2,x) - (n^2-3*n+2+(n^2-n-2)*x-(n^2-6*n+7)*x^2-(n^2-n-4)*x^3-(3*n-5)*x^4-2*x^5)*d/dx P(n-2,x) + (2*n^2-5*n+2+(2*n^2-4*n+4)*x+(5*n-11)*x^2+2*x^3)*P(n-1,x) + (n-2+2*x-(n-2)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 5*n^2 + 8*n - 4 + (2*n^3 - 9*n^2 + 9*n)*x + (n^3 - n^2 - 9*n + 11)*x^2 + (3*n^2 - 11*n + 12)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - (n^2 - 3*n + 2 + (n^2 - n - 2)*x - (n^2 - 6*n + 7)*x^2 - (n^2 - n - 4)*x^3 - (3*n - 5)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - 5*n + 2 + (2*n^2 - 4*n + 4)*x + (5*n - 11)*x^2 + 2*x^3)*P[n - 1, x] + (n - 2 + 2*x - (n - 2)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 2 + 2*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; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 5*n^2 + 8*n - 4 + (2*n^3 - 9*n^2 + 9*n)*x + (n^3 - n^2 - 9*n + 11)*x^2 + (3*n^2 - 11*n + 12)*x^3 + (2*n - 4)*x^4)*v[i - 2] - (n^2 - 3*n + 2 + (n^2 - n - 2)*x - (n^2 - 6*n + 7)*x^2 - (n^2 - n - 4)*x^3 - (3*n - 5)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - 5*n + 2 + (2*n^2 - 4*n + 4)*x + (5*n - 11)*x^2 + 2*x^3)*v[i - 1] + (n - 2 + 2*x - (n - 2)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 2 + 2*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-7b6a59413dc1b477","steps":["x \u21a6 x+1","D+1"],"edges":["e-121b28b1325b14d7","e-09db1f0bc490e140"],"text":"A102365 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A102365 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}