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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":"32b6a4368f0eed9cf2fe3f520333823ccb0ff4e8b4334c7aa253ef8b91bd14b5","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-a8ddb226edd55d5f","role":"sequence","name":"basis_map_A013611 of Pascal-(1,2,1) array read by antidiagonals.","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,      32;\n  3:     96,     320,      192;\n  4:    352,    1856,     2688,     1024;\n  5:   1200,    8704,    20736,    18432,     5120;\n  6:   3936,   36352,   121344,   178176,   112640,    24576;\n  7:  12544,  140800,   606720,  1261568,  1310720,   638976,   114688;\n  8:  39168,  517632,  2737152,  7409664, 10936320,  8699904,  3440640,   524288;\n  9: 120384, 1830912, 11483136, 38412288, 74158080, 83755008, 53673984, 17825792, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (6*n-6+(12*n^2-12*n)*x-(16*n-16)*x^2)*P(n,x) = -(18*n^2-78*n+48+(60*n^2-348*n+264)*x-(224*n-352)*x^2+128*x^3)*P(n-1,x) + (9*n-9+(72*n-84)*x+(156*n-252)*x^2+(48*n-256)*x^3-64*x^4)*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[(-(18*n^2 - 78*n + 48 + (60*n^2 - 348*n + 264)*x - (224*n - 352)*x^2 + 128*x^3)*P[n - 1, x] + (9*n - 9 + (72*n - 84)*x + (156*n - 252)*x^2 + (48*n - 256)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(6*n - 6 + (12*n^2 - 12*n)*x - (16*n - 16)*x^2)]];\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] = (-(18*n^2 - 78*n + 48 + (60*n^2 - 348*n + 264)*x - (224*n - 352)*x^2 + 128*x^3)*v[i - 1] + (9*n - 9 + (72*n - 84)*x + (156*n - 252)*x^2 + (48*n - 256)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(6*n - 6 + (12*n^2 - 12*n)*x - (16*n - 16)*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-f25cd955a876cde3","steps":["basis A013611","D"],"edges":["e-84c91f2034e460dd","e-31aff2a1df59ec8d"],"text":"A081577 \u2192 basis A013611 \u2192 D"},"headline":"A081577 \u2192 basis A013611 \u2192 D","chain":null}