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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":"b2513a2d15590ae10233ca7bd289b7487751721c696cab66a92d6ca080b7f483","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-20a6b63b1c7531be","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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:         11,          2;\n  2:        117,         44,          4;\n  3:       1215,        702,        132,         8;\n  4:      12393,       9720,       2808,       352,        16;\n  5:     124659,     123930,      48600,      9360,       880,       32;\n  6:    1240029,    1495908,     743580,    194400,     28080,     2112,      64;\n  7:   12223143,   17360406,   10471356,   3470040,    680400,    78624,    4928,    128;\n  8:  119574225,  195570288,  138883248,  55847232,  13880160,  2177280,  209664,  11264,   256;\n  9: 1162261467, 2152336050, 1760132592, 833299488, 251312544, 49968576, 6531840, 539136, 25344, 512;\n","recurrence_sentence":"The row polynomials satisfy (2*n+7+2*x)*P(n,x) = (18*n+81+(4*n+36)*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n + 81 + (4*n + 36)*x + 4*x^2)*P[n - 1, x])/(2*n + 7 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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