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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":"f339425a5ec2886f0e87776e562e9138e99482a4221b5a58a3f588a807d984e3","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-50226352e9cd0ce4","role":"sequence","name":"Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.","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:   0;\n  2:   1;\n  3:   4,     4;\n  4:  11,    36,     18;\n  5:  26,   196,    288,      96;\n  6:  57,   848,   2622,    2400,     600;\n  7: 120,  3228,  18120,   32976,   21600,    4320;\n  8: 247, 11364, 106374,  334000,  418140,  211680,   35280;\n  9: 502, 38044, 561864, 2804656, 5848840, 5495904, 2257920, 322560;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(2*n-2+(2*n^2-6*n+4)*x+(n^3-5*n^2+8*n-4)*x^2)*P(n-2,x) - ((n-1)*x+(n^2-3*n+2)*x^2)*d/dx P(n-2,x) + (3*n-5+(2*n^2-6*n+4)*x)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (2*n^2 - 6*n + 4)*x + (n^3 - 5*n^2 + 8*n - 4)*x^2)*P[n - 2, x] - ((n - 1)*x + (n^2 - 3*n + 2)*x^2)*D[P[n - 2, x], x] + (3*n - 5 + (2*n^2 - 6*n + 4)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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