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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":"04e09dba151a7aba24a20fb4dd56f1d88329c1f59e95dc74b57a32b09488b94f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2a657d152d421a92","role":"sequence","name":"Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  0;\n   2:  3;\n   3:  8,   24;\n   4: 15,  110,    165;\n   5: 24,  312,   1092,    1092;\n   6: 35,  700,   4200,    9520,    7140;\n   7: 48, 1360,  12240,   46512,   77520,    46512;\n   8: 63, 2394,  29925,  167580,  460845,   605682,   302841;\n   9: 80, 3920,  64680,  495880, 1983520,  4250400,  4604600,  1973400;\n  10: 99, 6072, 127512, 1275120, 6906900, 21312720, 37297260, 34337160, 12876435;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-6*n^2-4*n)*P(n,x) = (4*n^3+2*n^2-4*n-2+(18*n^3-15*n^2-6*n+3)*x)*P(n-1,x) + ((8*n^2-9*n+1)*x+(9*n^2-12*n+3)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 2*n^2 - 4*n - 2 + (18*n^3 - 15*n^2 - 6*n + 3)*x)*P[n - 1, x] + ((8*n^2 - 9*n + 1)*x + (9*n^2 - 12*n + 3)*x^2)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 - 4*n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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