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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":"7efcc913605d3edef9d58e69336ef94cc0b6b9d4b90ea3e65b00a6080fbf2170","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-677d34c7af372519","role":"sequence","name":"Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.","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:   0;\n  1:   1;\n  2:   3,     4;\n  3:   7,    24,     18;\n  4:  15,   100,    180,      96;\n  5:  31,   360,   1170,    1440,      600;\n  6:  63,  1204,   6300,   13440,    12600,     4320;\n  7: 127,  3864,  30618,  100800,   159600,   120960,    35280;\n  8: 255, 12100, 139860,  667296,  1587600,  1995840,  1270080,   322560;\n  9: 511, 37320, 613890, 4082400, 13696200, 25401600, 26460000, 14515200, 3265920;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+4*x+2*x^2)*P(n-2,x) - (x+2*x^2+x^3)*d/dx P(n-2,x) + (3+4*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(2 + 4*x + 2*x^2)*P[n - 2, x] - (x + 2*x^2 + x^3)*D[P[n - 2, x], x] + (3 + 4*x)*P[n - 1, x] + (x + x^2)*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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