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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":"e89032f94983e4da85fe89b7ece0a893953685f7f280e9ac03a2cdbf4d5599c7","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-0c0e99d973615582","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.","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:       3;\n  2:      30,        18;\n  3:     213,       426,         81;\n  4:    1308,      5964,       3924,         324;\n  5:    7431,     65292,      97938,       29724,        1215;\n  6:   40314,    620526,    1762884,     1241052,      201570,       4374;\n  7:  212505,   5391846,   26104635,    34806180,    13479615,    1275030,      15309;\n  8: 1099704,  44118072,  340082712,   765735960,   566804520,  132354216,    7697928,    52488;\n  9: 5623179, 346216440, 4053401028, 14317862568, 17897328210, 8106802056, 1211757540, 44985432, 177147;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = -(30*n+15+(24*n^2-24*n-18)*x+(8*n^3-12*n^2-2*n+3)*x^2)*P(n-2,x) - ((12*n+6)*x+(8*n^2-12*n-8)*x^2-(8*n^2-2)*x^3)*d/dx P(n-2,x) + (16*n-2+(8*n^2-8*n+2)*x)*P(n-1,x) + ((4*n-2)*x-(4*n-2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(30*n + 15 + (24*n^2 - 24*n - 18)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*P[n - 2, x] - ((12*n + 6)*x + (8*n^2 - 12*n - 8)*x^2 - (8*n^2 - 2)*x^3)*D[P[n - 2, x], x] + (16*n - 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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