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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":"bffb313a5f9e8a288aced471116bd2687509fba4fb660e3db37ffa109a6b4039","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-ba966decf74c4020","role":"sequence","name":"gamma_polynomial of Pascal-(1,9,1) array.","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:  9;\n  3: 18;\n  4: 27,  162;\n  5: 36,  486;\n  6: 45,  972,  2187;\n  7: 54, 1620,  8748;\n  8: 63, 2430, 21870,  26244;\n  9: 72, 3402, 43740, 131220;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n-1+(18*n-54)*x)*P(n-1,x) - (x+36*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 9.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 9;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (18*n - 54)*x)*P[n - 1, x] - (x + 36*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] = 9; for(n = 3, N, my(i = n + 1); v[i] = ((n - 1 + (18*n - 54)*x)*v[i - 1] - (x + 36*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-7ac21ea1b3b7e282","steps":["\u03b3","D"],"edges":["e-22858e6475490a6d","e-8093a5b2be32920d"],"text":"A143685 \u2192 \u03b3 \u2192 D"},"headline":"A143685 \u2192 \u03b3 \u2192 D","chain":null}