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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":"9ff358278302606dba268cd50cce4c0d2ca60a1a94f6beaf1b491b47db3c6250","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:   1;\n  1:   1;\n  2:   2,     1;\n  3:   5,     8,      2;\n  4:  12,    47,     36,       6;\n  5:  27,   222,    386,     192,      24;\n  6:  58,   905,   3046,    3274,    1200,     120;\n  7: 121,  3348,  19734,   39016,   29844,    8640,     720;\n  8: 248, 11611, 112056,  369458,  501640,  295308,   70560,   5040;\n  9: 503, 38546, 580886, 2991944, 6550004, 6665672, 3173904, 645120, 40320;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = -(2*n-4+(2*n^2-9*n+10)*x+(n^3-7*n^2+16*n-12)*x^2)*P(n-2,x) - ((n-2)*x+(n^2-5*n+6)*x^2)*d/dx P(n-2,x) + (3*n-8+(2*n^2-9*n+9)*x)*P(n-1,x) + (n-3)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 5 + 8*x + 2*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[3, x_] := 5 + 8*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (2*n^2 - 9*n + 10)*x + (n^3 - 7*n^2 + 16*n - 12)*x^2)*P[n - 2, x] - ((n - 2)*x + (n^2 - 5*n + 6)*x^2)*D[P[n - 2, x], x] + (3*n - 8 + (2*n^2 - 9*n + 9)*x)*P[n - 1, x] + (n - 3)*x*D[P[n - 1, x], x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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