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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":"a889d7bf75d09ec8e116c936cc95a263809c67a009426f32d0c289ff0ff7cf03","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-c85abc56078671cc","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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:   3,    2;\n  2:   9,   14,     3;\n  3:  19,   68,    37,      4;\n  4:  33,  228,   266,     76,      5;\n  5:  51,  590,  1278,    756,    135,      6;\n  6:  73, 1282,  4565,   4884,   1771,    218,     7;\n  7:  99, 2464, 13195,  22880,  14729,   3640,   329,     8;\n  8: 129, 4328, 32676,  84552,  87230,  37752,  6804,   472,   9;\n  9: 163, 7098, 72012, 261528, 400010, 274924, 85884, 11832, 651, 10;\n","recurrence_sentence":"The row polynomials satisfy -(8*n^3-22*n^2+24*n-9-(2*n^3-19*n^2+48*n-27)*x-(4*n^2-15*n+6)*x^2-2*n*x^3)*P(n,x) = -(8*n^3-6*n^2+4*n-3+(22*n^3-37*n^2-44*n+49)*x-(6*n^3-51*n^2+99*n-61)*x^2-(12*n^2-41*n+25)*x^3-(6*n-2)*x^4)*P(n-1,x) - ((16*n^2-8*n-12)*x-(20*n^2-40*n+22)*x^2+(4*n^2-40*n+56)*x^3+(8*n-26)*x^4+4*x^5)*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[(-(8*n^3 - 6*n^2 + 4*n - 3 + (22*n^3 - 37*n^2 - 44*n + 49)*x - (6*n^3 - 51*n^2 + 99*n - 61)*x^2 - (12*n^2 - 41*n + 25)*x^3 - (6*n - 2)*x^4)*P[n - 1, x] - ((16*n^2 - 8*n - 12)*x - (20*n^2 - 40*n + 22)*x^2 + (4*n^2 - 40*n + 56)*x^3 + (8*n - 26)*x^4 + 4*x^5)*D[P[n - 1, x], x])/(-8*n^3 + 22*n^2 - 24*n + 9 + (2*n^3 - 19*n^2 + 48*n - 27)*x + (4*n^2 - 15*n + 6)*x^2 + 2*n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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