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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":"06d28e65567e26f7d9476cc50145f2099799ebdbd727b6314e7b555f955bb49f","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-f48bed27dbc29c5f","role":"sequence","name":"The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=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:  8,   6,    1;\n  1: 12,  18,    8,    1;\n  2: 16,  38,   31,   10,    1;\n  3: 20,  66,   82,   47,   12,    1;\n  4: 24, 102,  173,  148,   66,   14,    1;\n  5: 28, 146,  316,  365,  240,   88,   16,    1;\n  6: 32, 198,  523,  766,  675,  362,  113,   18,   1;\n  7: 36, 258,  806, 1435, 1596, 1141,  518,  141,  20,   1;\n  8: 40, 326, 1177, 2472, 3332, 2996, 1806,  712, 172,  22,  1;\n  9: 44, 402, 1648, 3993, 6336, 6888, 5208, 2718, 948, 206, 24, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n+2+(n+4)*x+x^2)*P(n,x) = (2*n+4+(3*n+9)*x+(n+6)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 6*x + x^2.","mathematica":"P[0, x_] := 8 + 6*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + (3*n + 9)*x + (n + 6)*x^2 + x^3)*P[n - 1, x])/(2*n + 2 + (n + 4)*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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