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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":"df47ef00e5c2e45192b28d03a8d9947a9d0bd9b76e3b952a2f4ba01d631a8d55","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-0552ada0abe44307","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,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:    0;\n  1:    4;\n  2:   16,    32;\n  3:   40,   192,    192;\n  4:   88,   672,   1536,    1024;\n  5:  180,  1888,   6912,   10240,    5120;\n  6:  352,  4704,  23808,   56320,   61440,    24576;\n  7:  668, 10848,  70272,  230400,  399360,   344064,   114688;\n  8: 1240, 23712, 187392,  788480, 1894400,  2580480,  1835008,  524288;\n  9: 2264, 49824, 465024, 2393088, 7388160, 13934592, 15597568, 9437184, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-16*n^2-4*n+16-(32*n^2-32)*x)*P(n,x) = -(8*n^3-88*n^2+232*n-80-(128*n^2-576*n+64)*x+384*n*x^2)*P(n-1,x) + (5*n^2-25*n+10+(24*n^2-160*n+48)*x+(16*n^2-272*n+32)*x^2-128*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 88*n^2 + 232*n - 80 - (128*n^2 - 576*n + 64)*x + 384*n*x^2)*P[n - 1, x] + (5*n^2 - 25*n + 10 + (24*n^2 - 160*n + 48)*x + (16*n^2 - 272*n + 32)*x^2 - 128*n*x^3)*D[P[n - 1, x], x])/(4*n^3 - 16*n^2 - 4*n + 16 - (32*n^2 - 32)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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