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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":"45fc534dffc82fb2c1ac56365a540e7287a0e9e9c6a7b75fe9179f9cfb644caa","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-bb74f03367de3ec8","role":"sequence","name":"Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.","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:    3;\n  2:   15,    12;\n  3:   45,    90,     30;\n  4:  105,   378,    315,     60;\n  5:  210,  1176,   1764,    840,    105;\n  6:  378,  3024,   7056,   6048,   1890,    168;\n  7:  630,  6804,  22680,  30240,  17010,   3780,   252;\n  8:  990, 13860,  62370, 118800, 103950,  41580,  6930,   360;\n  9: 1485, 26136, 152460, 392040, 490050, 304920, 91476, 11880, 495;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n-6)*P(n,x) = (n^2+9*n+18+(3*n^2+9*n+2)*x)*P(n-1,x) + ((2*n+5)*x-(2*n+5)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 9*n + 18 + (3*n^2 + 9*n + 2)*x)*P[n - 1, x] + ((2*n + 5)*x - (2*n + 5)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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