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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":"1be89ce4ecd09418c40a1af857017cca854ee5bdb8ca1a3adc6728b4b9bc089f","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-1fe6eede339bc6ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/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:     9,     1;\n  3:    27,     6;\n  4:    80,    27,    1;\n  5:   234,   108,    9;\n  6:   677,   402,   54,   1;\n  7:  1941,  1422,  270,  12;\n  8:  5523,  4841, 1209,  90,  1;\n  9: 15615, 15996, 5013, 540, 15;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3+6*n^2-6*n+(12*n^2-27*n)*x+12*n*x^2)*P(n,x) = (7*n^3+25*n^2-25*n+2+(2*n^3+30*n^2-65*n-12)*x+(8*n^2+10*n+18)*x^2+(8*n-8)*x^3)*P(n-1,x) + (5*n^2+20*n-15-(n^2-16*n+2)*x-(4*n^2+20*n-33)*x^2-16*n*x^3-16*x^4)*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[((7*n^3 + 25*n^2 - 25*n + 2 + (2*n^3 + 30*n^2 - 65*n - 12)*x + (8*n^2 + 10*n + 18)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] + (5*n^2 + 20*n - 15 - (n^2 - 16*n + 2)*x - (4*n^2 + 20*n - 33)*x^2 - 16*n*x^3 - 16*x^4)*D[P[n - 1, x], x])/(3*n^3 + 6*n^2 - 6*n + (12*n^2 - 27*n)*x + 12*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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