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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":"79dfdf3d91104356a4a874fe7ecd09375115cea703e51e68d23524199d40e72d","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-d6b7687ba0ff59dd","role":"sequence","name":"Riordan array ((1+x)/(1-2x),x(1+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:    4,     1;\n  2:   12,     8,     1;\n  3:   33,    39,    12,     1;\n  4:   84,   150,    81,    16,     1;\n  5:  204,   498,   405,   138,    20,     1;\n  6:  480,  1500,  1674,   852,   210,    24,    1;\n  7: 1104,  4224,  6075,  4233,  1545,   297,   28,   1;\n  8: 2496, 11328, 20088, 18060,  8985,  2538,  399,  32,  1;\n  9: 5568, 29280, 61992, 68916, 44178, 16929, 3885, 516, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-20*n^2+18*n-4-(3*n^3-10*n^2+49*n-42)*x-(4*n^2-12*n-8)*x^2-(n^2+3*n+6)*x^3-(n+1)*x^4)*P(n,x) = -(12*n^3-28*n^2+16+(12*n^3-40*n^2-76*n+104)*x+(16*n^2-96*n+144)*x^2+(4*n^2-28)*x^3+(2*n-20)*x^4-2*x^5)*P(n-1,x) + ((12*n^2-4*n-8)*x+(24*n^2+16*n-48)*x^2+(3*n^2+51*n-42)*x^3+(14*n+44)*x^4+(n+14)*x^5+x^6)*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[(-(12*n^3 - 28*n^2 + 16 + (12*n^3 - 40*n^2 - 76*n + 104)*x + (16*n^2 - 96*n + 144)*x^2 + (4*n^2 - 28)*x^3 + (2*n - 20)*x^4 - 2*x^5)*P[n - 1, x] + ((12*n^2 - 4*n - 8)*x + (24*n^2 + 16*n - 48)*x^2 + (3*n^2 + 51*n - 42)*x^3 + (14*n + 44)*x^4 + (n + 14)*x^5 + x^6)*D[P[n - 1, x], x])/(-6*n^3 + 20*n^2 - 18*n + 4 + (3*n^3 - 10*n^2 + 49*n - 42)*x + (4*n^2 - 12*n - 8)*x^2 + (n^2 + 3*n + 6)*x^3 + (n + 1)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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