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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":"e6a41f0c2f5a6c9d97de13b9933e054f3244257da13afdb45381ea555b94ad4a","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-4ca874876022e136","role":"sequence","name":"Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","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:   9,   20,     6;\n  3:  16,   75,    60,     10;\n  4:  25,  204,   330,    140,     15;\n  5:  36,  455,  1260,   1050,    280,     21;\n  6:  49,  888,  3780,   5320,   2730,    504,     28;\n  7:  64, 1575,  9576,  20580,  17640,   6174,    840,    36;\n  8:  81, 2600, 21420,  65520,  85260,  49392,  12600,  1320,   45;\n  9: 100, 4059, 43560, 180180, 332640, 291060, 121968, 23760, 1980, 55;\n","recurrence_sentence":"The row polynomials satisfy (n^4+2*n^3+(2*n^3+2*n^2-4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (n^4+4*n^3+5*n^2+2*n+(3*n^4+8*n^3+2*n^2-4*n)*x+(6*n^3+6*n^2+2*n-2)*x^2+(3*n^2+4*n+2)*x^3)*P(n-1,x) + ((2*n^3+5*n^2+2*n)*x-(2*n^3+n^2-2*n)*x^2-(4*n^2+2*n-2)*x^3-(2*n+2)*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[((n^4 + 4*n^3 + 5*n^2 + 2*n + (3*n^4 + 8*n^3 + 2*n^2 - 4*n)*x + (6*n^3 + 6*n^2 + 2*n - 2)*x^2 + (3*n^2 + 4*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 + 5*n^2 + 2*n)*x - (2*n^3 + n^2 - 2*n)*x^2 - (4*n^2 + 2*n - 2)*x^3 - (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 + (2*n^3 + 2*n^2 - 4*n)*x + (n^2 + 2*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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