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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":"affb53c818bd30daea1dade0791f96ebdfb75cd01b1b716ec0224dc801c6c5c7","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-35273d9764d27bc4","role":"sequence","name":"basis_map_A013611 of 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:      4;\n  2:     32,      32;\n  3:    176,     384,      192;\n  4:    832,    2880,     3072,     1024;\n  5:   3632,   17408,    29184,    20480,      5120;\n  6:  15104,   92928,   215040,   235520,    122880,     24576;\n  7:  60800,  457728,  1359360,  2048000,   1658880,    688128,   114688;\n  8: 239104, 2131456,  7753728, 14970880,  16629760,  10665984,  3670016,   524288;\n  9: 923968, 9523200, 41084928, 97140736, 138055680, 121110528, 64225280, 18874368, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n+4+(2*n+4)*x)*P(n-1,x) + (1+4*x+2*x^2)*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[((2*n + 4 + (2*n + 4)*x)*P[n - 1, x] + (1 + 4*x + 2*x^2)*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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