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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":"39f3b041c8aca65524a339b6bd286fde7bf651842b33efef360ae525f5132827","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-170093e6e241015d","role":"sequence","name":"Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       0;\n   2:       3;\n   3:      21,        36;\n   4:     117,       540,        486;\n   5:     609,      5724,      12636,        7776;\n   6:    3093,     53100,     216270,      311040,       145800;\n   7:   15561,    461916,    3095820,     7814880,      8310600,      3149280;\n   8:   77997,   3882060,   40289886,   158941440,    287809200,    242494560,     77157360;\n   9:  390369,  31992444,  495596556,  2869818336,   7872908400,  11066569920,   7715736000,   2116316160;\n  10: 1952613, 260623980, 5883460110, 48107001600, 187648536600, 391386219840, 448592891040, 266655836160, 64283103360;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(10+24*x+18*x^2)*P(n-2,x) - (6*x+15*x^2+9*x^3)*d/dx P(n-2,x) + (7+12*x)*P(n-1,x) + (3*x+3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-(10 + 24*x + 18*x^2)*P[n - 2, x] - (6*x + 15*x^2 + 9*x^3)*D[P[n - 2, x], x] + (7 + 12*x)*P[n - 1, x] + (3*x + 3*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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