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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":"16725ee9d5b29de2731fe9d56a1257e4803c36f545bb02520c32db28d6e8bb27","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-ff9680a2cfd23e93","role":"sequence","name":"Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.","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:  1;\n  1:  3;\n  2:  5,   4;\n  3:  7,  16;\n  4:  9,  36,   12;\n  5: 11,  64,   60;\n  6: 13, 100,  168,   32;\n  7: 15, 144,  360,  192;\n  8: 17, 196,  660,  640,  80;\n  9: 19, 256, 1092, 1600, 560;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^2-n-(2*n^2+8*n)*x)*P(n,x) = -(2*n^2+n+(6*n^2-20*n-16)*x-(8*n^2+8*n-16)*x^2)*P(n-1,x) + ((2*n+1)*x+(14*n+4)*x^2-(16*n+32)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + n + (6*n^2 - 20*n - 16)*x - (8*n^2 + 8*n - 16)*x^2)*P[n - 1, x] + ((2*n + 1)*x + (14*n + 4)*x^2 - (16*n + 32)*x^3)*D[P[n - 1, x], x])/(-2*n^2 + n + (2*n^2 + 8*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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