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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":"70c04f7b5148ea855e4c345c8c62f6188bec47662369c95f2ed16b02eda0210a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5fa21c8779d2bdd1","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:  4,    3;\n  2:  8,   13,     3;\n  3: 13,   56,    31,      3;\n  4: 19,  156,   189,     52,      3;\n  5: 26,  343,   882,    485,     76,      3;\n  6: 34,  653,  2959,   3211,   1007,    103,     3;\n  7: 43, 1128,  7938,  16037,   9168,   1827,   133,    3;\n  8: 53, 1816, 18276,  61018,  62735,  21978,  3026,  166,   3;\n  9: 64, 2771, 37684, 189122, 331858, 197729, 46426, 4694, 202, 3;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(n^3+n^2-2*n)*x^2*P(n-4,x) - ((2*n^2+4*n)*x+(2*n^2+3*n)*x^2)*P(n-3,x) - (n+2-(2*n^2+2*n-2)*x+(n+1)*x^2)*P(n-2,x) + (2*n+3+(2*n+2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 3*x, P(2,x) = 8 + 13*x + 3*x^2, P(3,x) = 13 + 56*x + 31*x^2 + 3*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[2, x_] := 8 + 13*x + 3*x^2;\nP[3, x_] := 13 + 56*x + 31*x^2 + 3*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + n^2 - 2*n)*x^2*P[n - 4, x] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*P[n - 3, x] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*P[n - 2, x] + (2*n + 3 + (2*n + 2)*x)*P[n - 1, x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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