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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":"db884bebb86eed58f6178e7c34eed7b7bf872ea74a749305b1389643fe794a2b","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-d5c14b9fec875ecc","role":"sequence","name":"Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:    1;\n  2:    5,     2;\n  3:   16,    16,     3;\n  4:   44,    74,    33,     4;\n  5:  112,   268,   201,    56,     5;\n  6:  272,   848,   915,   424,    85,    6;\n  7:  640,  2464,  3504,  2336,   770,  120,    7;\n  8: 1472,  6752, 11976, 10564,  4990, 1266,  161,   8;\n  9: 3328, 17728, 37776, 41768, 26105, 9444, 1939, 208, 9;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^2-14*n+8-(3*n^2+n-4)*x+(n-1)*x^2)*P(n,x) = -(12*n^2-16*n+4+(12*n^2-50*n+10)*x-(13*n-11)*x^2+2*x^3)*P(n-1,x) + ((12*n-4)*x+(24*n-12)*x^2+(3*n-9)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n^2 - 16*n + 4 + (12*n^2 - 50*n + 10)*x - (13*n - 11)*x^2 + 2*x^3)*P[n - 1, x] + ((12*n - 4)*x + (24*n - 12)*x^2 + (3*n - 9)*x^3 - x^4)*D[P[n - 1, x], x])/(-6*n^2 + 14*n - 8 + (3*n^2 + n - 4)*x - (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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