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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":"a28d1fcea175e91870de95c695b9d7ef10aa99df3f10cc64404bfb7de8233c72","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-a7d7f0f34831fe03","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= n.","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:       1;\n   3:      10,       2;\n   4:      70,      42,        3;\n   5:     420,     504,      108,        4;\n   6:    2310,    4620,     1980,      220,       5;\n   7:   12012,   36036,    25740,     5720,     390,      6;\n   8:   60060,  252252,   270270,   100100,   13650,    630,     7;\n   9:  291720, 1633632,  2450448,  1361360,  309400,  28560,   952,    8;\n  10: 1385670, 9976824, 19953648, 15519504, 5290740, 813960, 54264, 1368, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n-2)*P(n,x) = (8*n^2-2+(6*n^2-16*n+8)*x)*P(n-1,x) + ((16*n-12)*x-(4*n-3)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 2 + (6*n^2 - 16*n + 8)*x)*P[n - 1, x] + ((16*n - 12)*x - (4*n - 3)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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