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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":"c713e721570f6f01ee05cb7495491497effc66a9b47f30c4d90f31d5a57d6691","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-81e6fb4f9f226d77","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(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:         13,          1;\n  2:        168,         26,          1;\n  3:       2160,        504,         39,         1;\n  4:      27648,       8640,       1008,        52,        1;\n  5:     352512,     138240,      21600,      1680,       65,       1;\n  6:    4478976,    2115072,     414720,     43200,     2520,      78,      1;\n  7:   56733696,   31352832,    7402752,    967680,    75600,    3528,     91,    1;\n  8:  716636160,  453869568,  125411328,  19740672,  1935360,  120960,   4704,  104,   1;\n  9: 9029615616, 6449725440, 2042413056, 376233984, 44416512, 3483648, 181440, 6048, 117, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+11+x)*P(n,x) = (12*n+144+(n+24)*x+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[((12*n + 144 + (n + 24)*x + x^2)*P[n - 1, x])/(n + 11 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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