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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":"a850ea851c7841deaef8124fa1abc2c3404dad63f00902f8e609761229946b07","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-dfab35176bd08e72","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:   7,    4;\n  1:  16,   22,     6;\n  2:  30,   72,    45,     8;\n  3:  50,  182,   192,    76,     10;\n  4:  77,  392,   612,   400,    115,    12;\n  5: 112,  756,  1620,  1540,    720,   162,    14;\n  6: 156, 1344,  3762,  4840,   3250,  1176,   217,    16;\n  7: 210, 2244,  7920, 13156,  11830,  6090,  1792,   280,   18;\n  8: 275, 3564, 15444, 32032,  36855, 25200, 10472,  2592,  351,  20;\n  9: 352, 5434, 28314, 71500, 101920, 88536, 48552, 16872, 3600, 430, 22;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^2+14*n+12-(4*n^2+10*n+6)*x)*P(n,x) = -(2*n^2+20*n+42-(3*n^2+8*n-3)*x-(2*n^2+10*n+12)*x^2)*P(n-1,x) - ((4*n+20)*x-(7*n+11)*x^2-(2*n+4)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 7 + 4*x.","mathematica":"P[0, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + 20*n + 42 - (3*n^2 + 8*n - 3)*x - (2*n^2 + 10*n + 12)*x^2)*P[n - 1, x] - ((4*n + 20)*x - (7*n + 11)*x^2 - (2*n + 4)*x^3)*D[P[n - 1, x], x])/(-2*n^2 - 14*n - 12 + (4*n^2 + 10*n + 6)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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