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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":"ad2169067a3cec9b175eab02b41ce62d287de48438fd9394eaddd4674b5c209d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f46f66d0d087b029","role":"sequence","name":"Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).","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:       8,        2;\n  1:      45,       36,         3;\n  2:     224,      336,        96,         4;\n  3:    1050,     2400,      1350,       200,         5;\n  4:    4752,    14850,     13200,      3960,       360,        6;\n  5:   21021,    84084,    105105,     50960,      9555,      588,        7;\n  6:   91520,   448448,    733824,    509600,    156800,    20160,      896,       8;\n  7:  393822,  2291328,   4678128,   4318272,   1927800,   411264,    38556,    1296,      9;\n  8: 1679600, 11337300,  27907200,  32558400,  19535040,  6104700,   957600,   68400,   1800,   10;\n  9: 7113106, 54716200, 158286150, 225118080, 172356030, 72997848, 16897650, 2032800, 114345, 2420, 11;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+14*n^2+16*n+6)*P(n,x) = (15*n^3+65*n^2+96*n+48+(12*n^3+43*n^2+45*n+12)*x)*P(n-1,x) + (n^2+2*n+1+(7*n^2+23*n+18)*x-(8*n^2+25*n+19)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8 + 2*x.","mathematica":"P[0, x_] := 8 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n^3 + 65*n^2 + 96*n + 48 + (12*n^3 + 43*n^2 + 45*n + 12)*x)*P[n - 1, x] + (n^2 + 2*n + 1 + (7*n^2 + 23*n + 18)*x - (8*n^2 + 25*n + 19)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 14*n^2 + 16*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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