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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":"29497fc5c8639ecb458a17d6374ea7a0a2570f50d69478dbd1cf094bcd8fcbef","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-91d76a88748fc9ba","role":"sequence","name":"Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.","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:      14,        4;\n  1:      72,       54,         6;\n  2:     330,      440,       132,         8;\n  3:    1430,     2860,      1560,       260,        10;\n  4:    6006,    16380,     13650,      4200,       450,       12;\n  5:   24752,    86632,     99960,     47600,      9520,      714,       14;\n  6:  100776,   434112,    651168,    434112,    135660,    19152,     1064,      16;\n  7:  406980,  2093040,   3907008,   3418632,   1508220,   335160,    35280,    1512,    18;\n  8: 1634380,  9806280,  22064130,  24227280,  14132580,  4462920,   743820,   60720,  2070,   20;\n  9: 6537520, 44945450, 118973250, 158631000, 116886000, 49092120, 11688600, 1518000, 99000, 2750, 22;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+34*n^2+86*n+56)*P(n,x) = (15*n^3+130*n^2+387*n+392+(12*n^3+86*n^2+186*n+112)*x)*P(n-1,x) + (n^2+3*n+2+(7*n^2+45*n+68)*x-(8*n^2+48*n+70)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 4*x.","mathematica":"P[0, x_] := 14 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n^3 + 130*n^2 + 387*n + 392 + (12*n^3 + 86*n^2 + 186*n + 112)*x)*P[n - 1, x] + (n^2 + 3*n + 2 + (7*n^2 + 45*n + 68)*x - (8*n^2 + 48*n + 70)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 34*n^2 + 86*n + 56)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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