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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":"c5147576b1e52d74a33362bd8deb86aacef22354527937c664693cff22c8412b","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-8b17fbdec05b5547","role":"sequence","name":"Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.","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:       4;\n  3:      18,       2;\n  4:      96,      18;\n  5:     600,     144,     3;\n  6:    4320,    1200,    48;\n  7:   35280,   10800,   600,    4;\n  8:  322560,  105840,  7200,  100;\n  9: 3265920, 1128960, 88200, 1800, 5;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*n-2)*x*P(n-3,x) - (n^2-n-2*x)*P(n-2,x) + 2*n*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4, P(3,x) = 18 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 4;\nP[3, x_] := 18 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*n - 2)*x*P[n - 3, x] - (n^2 - n - 2*x)*P[n - 2, x] + 2*n*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 4; v[4] = 18 + 2*x; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*n - 2)*x*v[i - 3] - (n^2 - n - 2*x)*v[i - 2] + 2*n*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-8b17fbdec05b5547","steps":["D"],"edges":["e-2fa13fafcaee11c4"],"text":"A084950 \u2192 D"},"headline":"A084950 \u2192 D","chain":null}