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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":"6327c5ce28385d603e31c83ef6f5f734cf8d4885129dad1692f8d506e82d4599","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-373413a99f5cc9e6","role":"sequence","name":"veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","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:     16;\n  5:    160,     2;\n  6:   1280,    48;\n  7:   8960,   672,    3;\n  8:  57344,  7168,   96;\n  9: 344064, 64512, 1728, 4;\n","recurrence_sentence":"The row polynomials satisfy (4*n-12)*P(n,x) = (16*n+(n-4)*x)*P(n-1,x) + (32*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[((16*n + (n - 4)*x)*P[n - 1, x] + (32*x - 2*x^2)*D[P[n - 1, x], x])/(4*n - 12)]];\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] = ((16*n + (n - 4)*x)*v[i - 1] + (32*x - 2*x^2)*deriv(v[i - 1], x))/(4*n - 12)); 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-3dcaebc50e901d48","steps":["Veronese 2 (j=1)","D"],"edges":["e-4a3fbc77a3e37655","e-9089e86c760261df"],"text":"A038231 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A038231 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}