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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":"4df6ad8d1ea355197a27c5eb749ee49dd7213edd518964e67ceb0c1c0977be3f","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-e7968148eadb4da9","role":"sequence","name":"basis_map_A007318 of Triangular array read by rows: 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:    1;\n  1:    5,    1;\n  2:    9,    2;\n  3:   28,   11,    1;\n  4:   62,   28,    3;\n  5:  165,   98,   18,   1;\n  6:  391,  266,   58,   4;\n  7:  983,  793,  225,  26,  1;\n  8: 2373, 2156,  711, 100,  5;\n  9: 5830, 5993, 2338, 425, 35, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3+13*n^2+33*n+21+(4*n^2+23*n+19)*x+(4*n+4)*x^2)*P(n,x) = (7*n^3+86*n^2+226*n+21+(2*n^3+52*n^2+225*n+19)*x+(8*n^2+74*n+4)*x^2+8*n*x^3)*P(n-1,x) - (39*n^2+546*n+1794+(25*n^2+506*n+2203)*x+(4*n^2+156*n+1015)*x^2+(16*n+208)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n^3 + 86*n^2 + 226*n + 21 + (2*n^3 + 52*n^2 + 225*n + 19)*x + (8*n^2 + 74*n + 4)*x^2 + 8*n*x^3)*P[n - 1, x] - (39*n^2 + 546*n + 1794 + (25*n^2 + 506*n + 2203)*x + (4*n^2 + 156*n + 1015)*x^2 + (16*n + 208)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + 13*n^2 + 33*n + 21 + (4*n^2 + 23*n + 19)*x + (4*n + 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((7*n^3 + 86*n^2 + 226*n + 21 + (2*n^3 + 52*n^2 + 225*n + 19)*x + (8*n^2 + 74*n + 4)*x^2 + 8*n*x^3)*v[i - 1] - (39*n^2 + 546*n + 1794 + (25*n^2 + 506*n + 2203)*x + (4*n^2 + 156*n + 1015)*x^2 + (16*n + 208)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + 13*n^2 + 33*n + 21 + (4*n^2 + 23*n + 19)*x + (4*n + 4)*x^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-ac04e52510a48bb5","steps":["basis A007318","D+1"],"edges":["e-16890f4567488209","e-af478967a03d4c17"],"text":"A249139 \u2192 basis A007318 \u2192 D+1"},"headline":"A249139 \u2192 basis A007318 \u2192 D+1","chain":null}