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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":"1e02bae2446e4a1a610a1d35e65ac9d2fb7f29c78a9593a97395569074f12e7e","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:    0;\n  1:    2;\n  2:   10,     8;\n  3:   32,    64,     24;\n  4:   88,   296,    264,     64;\n  5:  224,  1072,   1608,    896,    160;\n  6:  544,  3392,   7320,   6784,   2720,    384;\n  7: 1280,  9856,  28032,  37376,  24640,   7680,    896;\n  8: 2944, 27008,  95808, 169024, 159680,  81024,  20608,  2048;\n  9: 6656, 70912, 302208, 668288, 835360, 604416, 248192, 53248, 4608;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^2-7*n+4-(3*n^2+n-4)*x+(2*n-2)*x^2)*P(n,x) = -(6*n^2-8*n+2+(12*n^2-50*n+10)*x-(26*n-22)*x^2+8*x^3)*P(n-1,x) + ((6*n-2)*x+(24*n-12)*x^2+(6*n-18)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 - 8*n + 2 + (12*n^2 - 50*n + 10)*x - (26*n - 22)*x^2 + 8*x^3)*P[n - 1, x] + ((6*n - 2)*x + (24*n - 12)*x^2 + (6*n - 18)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(-3*n^2 + 7*n - 4 + (3*n^2 + n - 4)*x - (2*n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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