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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":"8e45bb991d3c77e21d5371efdee02379a86f1779d65f4e8843e870f32e500abc","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-8ce61d302cec295b","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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:  2;\n  2:  6,    8;\n  3: 12,   48,    24;\n  4: 20,  160,   240,     64;\n  5: 30,  400,  1200,    960,    160;\n  6: 42,  840,  4200,   6720,   3360,    384;\n  7: 56, 1568, 11760,  31360,  31360,  10752,    896;\n  8: 72, 2688, 28224, 112896, 188160, 129024,  32256,  2048;\n  9: 90, 4320, 60480, 338688, 846720, 967680, 483840, 92160, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n-2)*P(n,x) = (n^2+3*n+2+(6*n^2-2*n-4)*x)*P(n-1,x) + ((2*n+1)*x-(4*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n + 2 + (6*n^2 - 2*n - 4)*x)*P[n - 1, x] + ((2*n + 1)*x - (4*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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