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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":"a89550906d3b23e5245ca973748f2ec2b1423e6801601781ecbd77269f0ce58c","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-966c4f25f43e716e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:       3;\n  2:      36,       18;\n  3:     240,      432,        81;\n  4:    1344,     4320,      3240,       324;\n  5:    6912,    32256,     45360,     19440,      1215;\n  6:   33792,   207360,    435456,    362880,    102060,      4374;\n  7:  159744,  1216512,   3421440,   4354560,   2449440,    489888,    15309;\n  8:  737280,  6709248,  23721984,  41057280,  35925120,  14696640,  2204496,   52488;\n  9: 3342336, 35389440, 150958080, 332107776, 400308480, 258660864, 80831520, 9447840, 177147;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (8*n^2-12*n+4+(18*n^2-39*n+12)*x)*P(n-1,x) - ((4*n-2)*x+(12*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 12*n + 4 + (18*n^2 - 39*n + 12)*x)*P[n - 1, x] - ((4*n - 2)*x + (12*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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