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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":"125884effc736ae63d3accf321a2819fd6902a71ccd009eb63c69a2cbae492f6","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:       0;\n  1:       0;\n  2:       1;\n  3:       7,        2;\n  4:      51,       36,        3;\n  5:     431,      492,      123,        4;\n  6:    4208,     6646,     2964,      352,       5;\n  7:   46824,    94530,    61680,    14340,     915,      6;\n  8:  586141,  1441488,  1245435,   447340,   61125,   2244,    7;\n  9: 8161237, 23670236, 25524327, 12573460, 2798045, 240288, 5299, 8;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n^3-4*n^2+6*n-3+(2*n^3-10*n^2+14*n-6)*x+(n^3-6*n^2+11*n-6)*x^2)*P(n-2,x) - (n^2-2*n+1+(n^2-3*n+2)*x-(n^2-2*n+1)*x^2-(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (2*n^2-4*n+1+(2*n^2-8*n+8)*x)*P(n-1,x) + (n-2-(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 6*n - 3 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*P[n - 2, x] - (n^2 - 2*n + 1 + (n^2 - 3*n + 2)*x - (n^2 - 2*n + 1)*x^2 - (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 4*n + 1 + (2*n^2 - 8*n + 8)*x)*P[n - 1, x] + (n - 2 - (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + 6*n - 3 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*v[i - 2] - (n^2 - 2*n + 1 + (n^2 - 3*n + 2)*x - (n^2 - 2*n + 1)*x^2 - (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 4*n + 1 + (2*n^2 - 8*n + 8)*x)*v[i - 1] + (n - 2 - (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 2)); 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"],"edges":["e-121b28b1325b14d7","e-1c9db1f2a16cee48"],"text":"A102365 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A102365 \u2192 x \u21a6 x+1 \u2192 D","chain":null}