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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":"b6f89a55ce2699e29bc68918cafeb9a8397a0d941071564564eeaca4af25cb6f","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-9b5c7e97e5191da6","role":"sequence","name":"Triangular array formed by the Mersenne numbers.","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:    1;\n  2:    6,     2;\n  3:   23,    18,     3;\n  4:   72,    96,    36,     4;\n  5:  201,   396,   246,    60,     5;\n  6:  522,  1398,  1260,   500,    90,     6;\n  7: 1291,  4446,  5385,  3060,   885,   126,    7;\n  8: 3084, 13124, 20304, 15360,  6300,  1428,  168,   8;\n  9: 7181, 36648, 69828, 67032, 36330, 11592, 2156, 216, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (3*n+6+(n+2)*x)*P(n-1,x) + (1+6*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 6 + (n + 2)*x)*P[n - 1, x] + (1 + 6*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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