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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":"89dc921dde0124afd54a6126223d0e1b98a16eb817dc615df09c370e8280a0e0","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-e75bc09e82ca4254","role":"sequence","name":"derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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: 2, 2;\n  2: 0, 0, 12, 4;\n  3: 0, 0,  0, 0, 30, 6;\n  4: 0, 0,  0, 0,  0, 0, 56, 8;\n  5: 0, 0,  0, 0,  0, 0,  0, 0, 90, 10;\n  6: 0, 0,  0, 0,  0, 0,  0, 0,  0,  0, 132, 12;\n  7: 0, 0,  0, 0,  0, 0,  0, 0,  0,  0,   0,  0, 182, 14;\n  8: 0, 0,  0, 0,  0, 0,  0, 0,  0,  0,   0,  0,   0,  0, 240, 16;\n  9: 0, 0,  0, 0,  0, 0,  0, 0,  0,  0,   0,  0,   0,  0,   0,  0, 306, 18;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-5*n+3+(n-1)*x)*P(n,x) = ((2*n^2-n)*x^2+n*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n^2 - n)*x^2 + n*x^3)*P[n - 1, x])/(2*n^2 - 5*n + 3 + (n - 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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