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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":"596916d643fed773644aaed9aae0bbd65efb56deb5fb456b31579a7063a034a9","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-10f70e3ea8b205cf","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(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,          2;\n  2:        165,         52,          4;\n  3:       2057,        990,        156,          8;\n  4:      25289,      16456,       3960,        416,        16;\n  5:     307461,     252890,      82280,      13200,      1040,        32;\n  6:    3704173,    3689532,    1517340,     329120,     39600,      2496,       64;\n  7:   44289025,   51858422,   25826724,    7080920,   1151920,    110880,     5824,    128;\n  8:  526153617,  708624400,  414867376,  137742528,  28323680,   3686144,   295680,  13312,   256;\n  9: 6216407549, 9470765106, 6377619600, 2489204256, 619841376, 101965248, 11058432, 760320, 29952, 512;\n","recurrence_sentence":"The row polynomials satisfy (2*n+9+2*x)*P(n,x) = (22*n+121+(4*n+44)*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[((22*n + 121 + (4*n + 44)*x + 4*x^2)*P[n - 1, x])/(2*n + 9 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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