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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":"ba6a69141acb941ee0e50aa780ee547b9ab3b0aff3035dabef15a2a3d2504c48","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-54b58a0ddb0e34fe","role":"sequence","name":"A decomposition of the double factorials A001147.","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:    0;\n  2:    2;\n  3:   10,       8;\n  4:   36,     120,       24;\n  5:  116,    1032,      888,       64;\n  6:  358,    7016,    15504,     5312,      160;\n  7: 1086,   42240,   192720,   171840,    28320,      384;\n  8: 3272,  237264,  1982640,  3602240,  1600320,   140928,    896;\n  9: 9832, 1277936, 18149232, 58855360, 53625920, 13354368, 671104, 2048;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(3*n-3+(4*n^2-16*n+12)*x+(4*n^3-24*n^2+44*n-24)*x^2)*P(n-2,x) - ((2*n-2)*x+(4*n^2-14*n+10)*x^2-(4*n^2-12*n+8)*x^3)*d/dx P(n-2,x) + (4*n-7+(4*n^2-16*n+16)*x)*P(n-1,x) + ((2*n-4)*x-(2*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (4*n^2 - 16*n + 12)*x + (4*n^3 - 24*n^2 + 44*n - 24)*x^2)*P[n - 2, x] - ((2*n - 2)*x + (4*n^2 - 14*n + 10)*x^2 - (4*n^2 - 12*n + 8)*x^3)*D[P[n - 2, x], x] + (4*n - 7 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] + ((2*n - 4)*x - (2*n - 4)*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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