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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":"9b201c6e6c676e5048d305a0385efdb851128bab472e00ccc40f200c49b32277","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-a679863398df5985","role":"sequence","name":"Triangle of coefficients in expansion of (14 + x)^n.","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:          15,           1;\n  2:         224,          30,          1;\n  3:        3332,         672,         45,         1;\n  4:       49392,       13328,       1344,        60,        1;\n  5:      729904,      246960,      33320,      2240,       75,       1;\n  6:    10756480,     4379424,     740880,     66640,     3360,      90,      1;\n  7:   158120256,    75295360,   15327984,   1728720,   116620,    4704,    105,    1;\n  8:  2319097088,  1264962048,  301181440,  40874624,  3457440,  186592,   6272,  120,   1;\n  9: 33943148288, 20871873792, 5692329216, 903544320, 91967904, 6223392, 279888, 8064, 135, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+13+x)*P(n,x) = (14*n+196+(n+28)*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[((14*n + 196 + (n + 28)*x + x^2)*P[n - 1, x])/(n + 13 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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