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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":"9e5459a32ee68fe44e27dbee7c39c6694b487b73fff3b34bbcbf7b5ad02fcadd","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-01512c1ccffbc04e","role":"sequence","name":"T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.","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: 1,     12;\n  3: 1,     60,       180;\n  4: 1,    252,      2520,        3360;\n  5: 1,   1020,     26460,      100800,        75600;\n  6: 1,   4092,    253440,     2106720,      4158000,      1995840;\n  7: 1,  16380,   2342340,    38438400,    151351200,    181621440,     60540480;\n  8: 1,  65532,  21326760,   658738080,   4648644000,  10534043520,   8475667200,   2075673600;\n  9: 1, 262140, 192923820, 10937908800, 131037706800, 501949768320, 734840346240, 423437414400, 79394515200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -12*x^2*P(n-2,x) - (4*x^2+8*x^3)*d/dx P(n-2,x) - 2*x^3*d^2/dx^2 P(n-2,x) + (1+12*x)*P(n-1,x) + (3*x+4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 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[-12*x^2*P[n - 2, x] - (4*x^2 + 8*x^3)*D[P[n - 2, x], x] - 2*x^3*D[P[n - 2, x], {x, 2}] + (1 + 12*x)*P[n - 1, x] + (3*x + 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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