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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":"95ae835a8c69ef2ae1b13b835e5d9d83727a7668cefdbad06a67bb89710566d8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f1c436fe49d0588b","role":"sequence","name":"basis_map_A008277 of Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.","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:          5,          1;\n  2:         36,         14,          1;\n  3:        336,        192,         27,          1;\n  4:       3840,       2880,        600,         44,         1;\n  5:      51840,      48000,      13200,       1440,        65,        1;\n  6:     806400,     887040,     302400,      43680,      2940,       90,      1;\n  7:   14192640,   18063360,    7338240,    1317120,    117600,     5376,    119,     1;\n  8:  278691840,  402554880,  189665280,   40642560,   4515840,   274176,   9072,   152,   1;\n  9: 6038323200, 9754214400, 5225472000, 1300561920, 172730880, 13063680, 574560, 14400, 189, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+3-x)*P(n,x) = (2*n^2+10*n+8-n*x-x^2)*P(n-1,x) + ((2*n+4)*x-2*x^2)*d/dx 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[((2*n^2 + 10*n + 8 - n*x - x^2)*P[n - 1, x] + ((2*n + 4)*x - 2*x^2)*D[P[n - 1, x], x])/(n + 3 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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