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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":"3ed55989ff538e64f2c501252bcaf4f99837b075759a5fcc9e67d44e2d9d4cd5","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-7c641d81e21514ce","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=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":[{"id":"l-14efd2944a1e73f7","kind":"fingerprint_related","node_a_id":"n-08469142fb9c5873","node_b_id":"n-6dbf222e2d76f379","fingerprint":"f6-4f4ca2dec4972ed6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f4ca2dec4972ed6"},"other_id":"n-08469142fb9c5873","other":{"id":"n-08469142fb9c5873","name":"one_plus_derivative of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-4f4ca2dec4972ed6","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["3","2"]],"row_sums_preview":["1","1","5","5","21","21","81","81"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["D+1"],"edges":["e-90e13dda23ca7fe5"],"text":"A188440 \u2192 D+1"},"headline":"A188440 \u2192 D+1","chain":null}}],"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,     6;\n  3:   27,    18;\n  4:   45,   108,    36;\n  5:  135,   324,   108;\n  6:  189,   810,   972,   216;\n  7:  567,  2430,  2916,   648;\n  8:  729,  4536,  9720,  7776, 1296;\n  9: 2187, 13608, 29160, 23328, 3888;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-2+(4*n-6)*x+4*x^2)*P(n,x) = -(2*n^3-13*n^2+19*n-2-(2*n^3-12*n^2+34*n-30)*x-(8*n^2-20*n+24)*x^2-(8*n-8)*x^3)*P(n-1,x) + (2*n^2-4*n+(2*n^2-2*n-8)*x-(4*n^2-4*n+12)*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[(-(2*n^3 - 13*n^2 + 19*n - 2 - (2*n^3 - 12*n^2 + 34*n - 30)*x - (8*n^2 - 20*n + 24)*x^2 - (8*n - 8)*x^3)*P[n - 1, x] + (2*n^2 - 4*n + (2*n^2 - 2*n - 8)*x - (4*n^2 - 4*n + 12)*x^2 - 16*n*x^3 - 16*x^4)*D[P[n - 1, x], x])/(n^2 - n - 2 + (4*n - 6)*x + 4*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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