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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":"b0ad25dfd7844eb2919d3c322d7f990cbe26afe0ffa40d85f1118c21cb661980","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-abf94b4038449932","role":"sequence","name":"veronese_section_2_1 of Cube of lower triangular normalized binomial matrix.","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:     0;\n  3:     1;\n  4:    12;\n  5:    90,     2;\n  6:   540,    36;\n  7:  2835,   378,   3;\n  8: 13608,  3024,  72;\n  9: 61236, 20412, 972, 4;\n","recurrence_sentence":"The row polynomials satisfy (3*n-9)*P(n,x) = (9*n+(n-4)*x)*P(n-1,x) + (18*x-2*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n + (n - 4)*x)*P[n - 1, x] + (18*x - 2*x^2)*D[P[n - 1, x], x])/(3*n - 9)]];\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] = 0; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = ((9*n + (n - 4)*x)*v[i - 1] + (18*x - 2*x^2)*deriv(v[i - 1], x))/(3*n - 9)); 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-8d887bfc40d036b9","steps":["Veronese 2 (j=1)","D"],"edges":["e-59cb5e4e13a9546c","e-8a78129ac7b2b23b"],"text":"A027465 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A027465 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}