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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":"55e5296160c76dadeded209318531b74cbeff7c20cc32cdafbfe8a572e8d5ea1","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-15cf3f3457596bec","role":"sequence","name":"veronese_section_2_1 of Triangle of coefficients in expansion of (1+3*x)^n.","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:   27;\n  4:  108;\n  5:  270,   486;\n  6:  540,  2916;\n  7:  945, 10206,   6561;\n  8: 1512, 27216,  52488;\n  9: 2268, 61236, 236196, 78732;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (n+(9*n-36)*x)*P(n-1,x) + (2*x-18*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) = 27.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 27;\nP[n_, x_] := P[n, x] = Expand[Together[((n + (9*n - 36)*x)*P[n - 1, x] + (2*x - 18*x^2)*D[P[n - 1, x], x])/(n - 3)]];\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] = 27; for(n = 4, N, my(i = n + 1); v[i] = ((n + (9*n - 36)*x)*v[i - 1] + (2*x - 18*x^2)*deriv(v[i - 1], x))/(n - 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-6a5af5e1168bb679","steps":["Veronese 2 (j=1)","D"],"edges":["e-a72a0a226c85ec5c","e-26993e9e81ecc84e"],"text":"A013610 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A013610 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}