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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":"96054ae84c14033fa41881a0fc8aabb82cb289ef685332a47c97165e3dea3c1b","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-54aa7a9d2edaadfb","role":"sequence","name":"A triangle of numbers related to triangle A030527.","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:          7,           1;\n  2:         60,          20,           1;\n  3:        612,         348,          39,          1;\n  4:       7224,        6120,        1140,         64,         1;\n  5:      96264,      112824,       30780,       2820,        95,        1;\n  6:    1419264,     2201472,      819504,     108360,      5880,      132,       1;\n  7:   22796928,    45505152,    22098384,    3937584,    304920,    10920,     175,     1;\n  8:  394377984,   994194432,   610860096,  140374080,  14503104,   734832,   18648,   224,   1;\n  9: 7285614336, 22885099776, 17405720640, 5003570880, 660623040, 44307648, 1580040, 29880, 279, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+13*n+36-(3*n+12)*x-4*x^2)*P(n-2,x) - ((5*n+20)*x-20*x^2)*d/dx P(n-2,x) + (2*n+14-3*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 13*n + 36 - (3*n + 12)*x - 4*x^2)*P[n - 2, x] - ((5*n + 20)*x - 20*x^2)*D[P[n - 2, x], x] + (2*n + 14 - 3*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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