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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":"e5c8ee116544edd02f54364733551cc8d4170a3cf2de2aafe11dd8544d772659","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-92be85f47bd42a26","role":"sequence","name":"Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     2;\n   2:    16,      8;\n   3:    76,     96,      20;\n   4:   256,    648,     336,      40;\n   5:   686,   2992,    3036,     896,      70;\n   6:  1568,  10608,   18304,   10400,    2016,     112;\n   7:  3192,  31104,   82680,   79040,   29160,    4032,     168;\n   8:  5952,  79152,  301920,  442000,  272000,   70992,    7392,    240;\n   9: 10362, 180576,  937992, 1963840, 1841100,  795872,  155496,  12672,   330;\n  10: 17072, 377784, 2568192, 7307552, 9767520, 6403152, 2059904, 313632, 20592, 440;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-7*n^2+12*n-7+(4*n^2-13*n+9)*x+(2*n-2)*x^2)*P(n,x) = (2*n^3+3*n^2+2*n+8+(6*n^3-11*n^2+15*n-22)*x+(12*n^2-27*n+18)*x^2+(6*n-4)*x^3)*P(n-1,x) + ((4*n^2-4*n+6)*x-(4*n^2-12*n+16)*x^2-(8*n-14)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 3*n^2 + 2*n + 8 + (6*n^3 - 11*n^2 + 15*n - 22)*x + (12*n^2 - 27*n + 18)*x^2 + (6*n - 4)*x^3)*P[n - 1, x] + ((4*n^2 - 4*n + 6)*x - (4*n^2 - 12*n + 16)*x^2 - (8*n - 14)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 7*n^2 + 12*n - 7 + (4*n^2 - 13*n + 9)*x + (2*n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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