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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":"35edde304b33abdd6214b7621b9ce89d1d9018e9e16760fe8be5eafa01bf78c0","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-d07b286c67e2d8fc","role":"sequence","name":"basis_map_A013610 of Pascal-(1,4,1) array.","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,       3;\n  2:     32,      42,        9;\n  3:    132,     360,      207,       27;\n  4:    560,    2280,     2520,      864,       81;\n  5:   2176,   12000,    21240,    13824,     3321,      243;\n  6:   8320,   57888,   141840,   147744,    66096,    12150,      729;\n  7:  30912,  260352,   822384,  1212624,   861516,   289656,    43011,    2187;\n  8: 113152, 1119360,  4338432,  8358336,  8457696,  4469256,  1195560,  148716,   6561;\n  9: 408320, 4642560, 21363840, 51107328, 68230512, 51302160, 21345120, 4723920, 505197, 19683;\n","recurrence_sentence":"The row polynomials satisfy (270*n^2-90*n+(405*n^3+135*n^2-90*n)*x+(1377*n^2-1269*n)*x^2+(243*n^2+972*n)*x^3+243*n*x^4)*P(n,x) = -(1350*n^3-3870*n^2-300*n+1920+(3645*n^3-7965*n^2-9090*n+10620)*x+(9558*n^2-22680*n+11232)*x^2+(2187*n^2+3483*n-12069)*x^3+(1458*n-6318)*x^4-729*x^5)*P(n-1,x) + (900*n^2+300*n-200+(5400*n^2+4860*n-780)*x+(8505*n^2+21735*n+3870)*x^2+(1215*n^2+32562*n+23679)*x^3+(9234*n+34425)*x^4+(729*n+9477)*x^5+729*x^6)*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[(-(1350*n^3 - 3870*n^2 - 300*n + 1920 + (3645*n^3 - 7965*n^2 - 9090*n + 10620)*x + (9558*n^2 - 22680*n + 11232)*x^2 + (2187*n^2 + 3483*n - 12069)*x^3 + (1458*n - 6318)*x^4 - 729*x^5)*P[n - 1, x] + (900*n^2 + 300*n - 200 + (5400*n^2 + 4860*n - 780)*x + (8505*n^2 + 21735*n + 3870)*x^2 + (1215*n^2 + 32562*n + 23679)*x^3 + (9234*n + 34425)*x^4 + (729*n + 9477)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(270*n^2 - 90*n + (405*n^3 + 135*n^2 - 90*n)*x + (1377*n^2 - 1269*n)*x^2 + (243*n^2 + 972*n)*x^3 + 243*n*x^4)]];\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] = (-(1350*n^3 - 3870*n^2 - 300*n + 1920 + (3645*n^3 - 7965*n^2 - 9090*n + 10620)*x + (9558*n^2 - 22680*n + 11232)*x^2 + (2187*n^2 + 3483*n - 12069)*x^3 + (1458*n - 6318)*x^4 - 729*x^5)*v[i - 1] + (900*n^2 + 300*n - 200 + (5400*n^2 + 4860*n - 780)*x + (8505*n^2 + 21735*n + 3870)*x^2 + (1215*n^2 + 32562*n + 23679)*x^3 + (9234*n + 34425)*x^4 + (729*n + 9477)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(270*n^2 - 90*n + (405*n^3 + 135*n^2 - 90*n)*x + (1377*n^2 - 1269*n)*x^2 + (243*n^2 + 972*n)*x^3 + 243*n*x^4)); 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-ed9434bfd889c409","steps":["basis A013610","D+1"],"edges":["e-14634eb9644b375c","e-868d25809df5682d"],"text":"A081579 \u2192 basis A013610 \u2192 D+1"},"headline":"A081579 \u2192 basis A013610 \u2192 D+1","chain":null}