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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":"899651c671ba5f5a70c90ae62a0b341caba8a05653ff2512c4d3b965819a26b6","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-7d781fe2b71cdf22","role":"sequence","name":"Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].","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:         2,          1;\n  2:         9,          8,         1;\n  3:        60,         75,        18,         1;\n  4:       525,        840,       294,        32,        1;\n  5:      5670,      11025,      5040,       810,       50,       1;\n  6:     72765,     166320,     93555,     19800,     1815,      72,      1;\n  7:   1081080,    2837835,   1891890,    495495,    60060,    3549,     98,     1;\n  8:  18243225,   54054000,  41621580,  12972960,  1951950,  152880,   6300,   128,   1;\n  9: 344594250, 1137161025, 992431440, 358378020, 64324260, 6265350, 342720, 10404, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-x)*P(n,x) = (2*n^2+n-1-n*x-x^2)*P(n-1,x) + ((2*n-2)*x-2*x^2)*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^2 + n - 1 - n*x - x^2)*P[n - 1, x] + ((2*n - 2)*x - 2*x^2)*D[P[n - 1, x], x])/(n - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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