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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":"dc625b7627bbd3d9fea447314686fef03f3716afc9b752d087fb1b6c6aa84f21","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-77f76afdd288bbc1","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 5).","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:          15,            2;\n  1:         175,           60,            3;\n  2:        1875,         1250,          150,            4;\n  3:       19375,        22500,         4875,          300,            5;\n  4:      196875,       376250,       131250,        14000,          525,           6;\n  5:     1984375,      6037500,      3189375,       525000,        33250,         840,         7;\n  6:    19921875,     94531250,     72843750,     17377500,      1653750,       69300,      1260,        8;\n  7:   199609375,   1457812500,   1598671875,    531562500,     71334375,     4410000,    131250,     1800,      9;\n  8:  1998046875,  22266406250,  34160156250,  15420625000,   2804484375,   239951250,  10395000,   231000,   2475,   10;\n  9: 19990234375, 337992187500, 716602734375, 431062500000, 103410312500, 11763675000, 695743125, 22275000, 383625, 3300, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(50+10*x+x^2)*P(n-2,x) - (25*x+5*x^2)*d/dx P(n-2,x) + (15+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 15 + 2*x, P(1,x) = 175 + 60*x + 3*x^2.","mathematica":"P[0, x_] := 15 + 2*x;\nP[1, x_] := 175 + 60*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(50 + 10*x + x^2)*P[n - 2, x] - (25*x + 5*x^2)*D[P[n - 2, x], x] + (15 + 2*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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