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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":"569a42fccd4169df6158d395b24fa5301a152e12d5093c1ef1cdda1c029688b5","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-20086f90792ae22f","role":"sequence","name":"basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:      4;\n  2:     24,      16;\n  3:    124,     160,      48;\n  4:    576,    1104,     672,     128;\n  5:   2532,    6432,    5904,    2304,     320;\n  6:  10728,   33936,   41664,   24704,    7040,     768;\n  7:  44300,  167712,  257904,  205568,   89280,   19968,   1792;\n  8: 179472,  791120, 1461984, 1464960,  857600,  292608,  53760,   4096;\n  9: 716468, 3603872, 7778256, 9392384, 6927680, 3190272, 894208, 139264, 9216;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(2*n-2)*x)*P(n,x) = -(7*n-20+(6*n-28)*x-8*x^2)*P(n-1,x) + (17+37*x+24*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(7*n - 20 + (6*n - 28)*x - 8*x^2)*P[n - 1, x] + (17 + 37*x + 24*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(7*n - 20 + (6*n - 28)*x - 8*x^2)*v[i - 1] + (17 + 37*x + 24*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x)); 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-e646c30d067149e7","steps":["basis A007318","D"],"edges":["e-4395b7c136181822","e-b575c0d832433dc6"],"text":"A249252 \u2192 basis A007318 \u2192 D"},"headline":"A249252 \u2192 basis A007318 \u2192 D","chain":null}