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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":"ce055dabc73c5daa257b1a0ff509f74a5cf2d56992f6b09aa583db3ca62234e4","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-2feb6dc01f424b0a","role":"sequence","name":"Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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;\n  2:  5,   12;\n  3:  9,   52,    72;\n  4: 14,  142,   462,    480;\n  5: 20,  310,  1740,   4176,    3600;\n  6: 27,  590,  4995,  20416,   40140,    30240;\n  7: 35, 1022, 12075,  73696,  244300,   415584,   282240;\n  8: 44, 1652, 25872, 218596, 1073380,  3054024,  4645872,  2903040;\n  9: 54, 2532, 50652, 563556, 3805830, 15934584, 40276152, 55998720, 32659200;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2+2*n)*x+(n^3+n^2)*x^2)*P(n-2,x) + (2*n+1+(2*n^2+2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*P[n - 2, x] + (2*n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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