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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":"ccb56755e121c920eb64a087588c1341a17ec0907eceb526bf36190752be1578","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-92be85f47bd42a26","role":"sequence","name":"Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-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:     8;\n   3:    56,     40;\n   4:   216,    432,     120;\n   5:   616,   2376,    1848,     280;\n   6:  1456,   9152,   13728,    5824,     560;\n   7:  3024,  28080,   68640,   56160,   15120,    1008;\n   8:  5712,  73440,  265200,  353600,  183600,   34272,    1680;\n   9: 10032, 170544,  852720, 1679600, 1421200,  511632,   70224,   2640;\n  10: 16632, 361152, 2387616, 6511680, 8139600, 4775232, 1264032, 133056, 3960;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-5*n+2)*P(n,x) = (2*n^2+5*n+2+(6*n^2-9*n-2)*x)*P(n-1,x) + (4*n*x-4*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 8.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 5*n + 2 + (6*n^2 - 9*n - 2)*x)*P[n - 1, x] + (4*n*x - 4*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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