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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":"e51671d54429010ad94107747997b2d189811d9959f585cfeb29b4e3db609e83","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cbd7c2e66114e926","role":"sequence","name":"Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-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:        4;\n  2:       24,        16;\n  3:      144,       192,        48;\n  4:      960,      1920,       960,       128;\n  5:     7200,     19200,     14400,      3840,       320;\n  6:    60480,    201600,    201600,     80640,     13440,      768;\n  7:   564480,   2257920,   2822400,   1505280,    376320,    43008,    1792;\n  8:  5806080,  27095040,  40642560,  27095040,   9031680,  1548288,  129024,   4096;\n  9: 65318400, 348364800, 609638400, 487710720, 203212800, 46448640, 5806080, 368640, 9216;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n^2+n+2*n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + 2*n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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