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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":"3cf073bd8b154fc184b7458ee9da9407daa29d65b2040a8fc80b54480c09e006","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1bffaa6198344f93","role":"sequence","name":"Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:    4,     2;\n  2:   14,    18,      4;\n  3:   40,    96,     56,      8;\n  4:  104,   384,    412,    152,      16;\n  5:  256,  1296,   2144,   1432,     384,     32;\n  6:  608,  3936,   9016,   9224,    4416,    928,     64;\n  7: 1408, 11136,  32960,  46720,   33984,  12608,   2176,   128;\n  8: 3200, 29952, 109312, 200960,  201936, 112960,  34112,  4992,   256;\n  9: 7168, 77568, 337664, 769024, 1002432, 771488, 348928, 88704, 11264, 512;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^3-5*n^2+2*n-(3*n^3-5*n^2+14*n)*x-(4*n^2-8*n)*x^2-2*n^2*x^3-2*n*x^4)*P(n,x) = -(6*n^3-4*n^2-2*n+(12*n^3-26*n^2-34*n)*x+(10*n^2-54*n+36)*x^2+(8*n^2-4*n-4)*x^3+(4*n-16)*x^4-4*x^5)*P(n-1,x) + ((6*n^2+2*n)*x+(24*n^2+20*n-4)*x^2+(6*n^2+54*n-4)*x^3+(28*n+48)*x^4+(4*n+28)*x^5+4*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[(-(6*n^3 - 4*n^2 - 2*n + (12*n^3 - 26*n^2 - 34*n)*x + (10*n^2 - 54*n + 36)*x^2 + (8*n^2 - 4*n - 4)*x^3 + (4*n - 16)*x^4 - 4*x^5)*P[n - 1, x] + ((6*n^2 + 2*n)*x + (24*n^2 + 20*n - 4)*x^2 + (6*n^2 + 54*n - 4)*x^3 + (28*n + 48)*x^4 + (4*n + 28)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(-3*n^3 + 5*n^2 - 2*n + (3*n^3 - 5*n^2 + 14*n)*x + (4*n^2 - 8*n)*x^2 + 2*n^2*x^3 + 2*n*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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