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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":"6de785ab4a13ebdf97994487c46e3ca0a0113ec5f4d81f5974f0b3c3b0f12c26","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-695d6cb541611b1e","role":"sequence","name":"gamma_polynomial of Pascal-(1,7,1) array.","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:  1;\n  2:  8,    7;\n  3: 15,   14;\n  4: 22,  119,    49;\n  5: 29,  322,   147;\n  6: 36,  623,  1323,   343;\n  7: 43, 1022,  4606,  1372;\n  8: 50, 1519, 11025, 13034,  2401;\n  9: 57, 2114, 21609, 54880, 12005;\n","recurrence_sentence":"The row polynomials satisfy (7*n^3-20*n^2+13*n+(28*n^2-69*n)*x+28*n*x^2)*P(n,x) = (7*n^3-13*n^2+6*n+(98*n^3-350*n^2+211*n)*x+(392*n^2-938*n+574)*x^2+(392*n-392)*x^3)*P(n-1,x) - ((7*n^2-6*n)*x+(196*n^2-140*n-13)*x^2+(784*n-336)*x^3+784*x^4)*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[((7*n^3 - 13*n^2 + 6*n + (98*n^3 - 350*n^2 + 211*n)*x + (392*n^2 - 938*n + 574)*x^2 + (392*n - 392)*x^3)*P[n - 1, x] - ((7*n^2 - 6*n)*x + (196*n^2 - 140*n - 13)*x^2 + (784*n - 336)*x^3 + 784*x^4)*D[P[n - 1, x], x])/(7*n^3 - 20*n^2 + 13*n + (28*n^2 - 69*n)*x + 28*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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