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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":"8838af9a20b9fc60328557329269336fdaacc5f04de8f6454082ab658250ae80","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-5c56442adc0376c2","role":"sequence","name":"Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.","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:           15,           2;\n   4:          255,          60,           3;\n   5:         5175,        1650,         150,          4;\n   6:       123795,       48300,        6075,        300,         5;\n   7:      3427515,     1551690,      242550,      16800,       525,       6;\n   8:    108046575,    54957420,    10089135,     877800,     38850,     840,      7;\n   9:   3824996175,  2139602850,   445644990,   45930780,   2573550,   79380,   1260,    8;\n  10: 150346471275, 91094503500, 21035203875, 2477355300, 165502575, 6501600, 148050, 1800, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-52*n+42+(8*n-14)*x+x^2)*P(n-2,x) - ((4*n-7)*x+x^2)*d/dx P(n-2,x) + (8*n-9+2*x)*P(n-1,x) + x*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[-(16*n^2 - 52*n + 42 + (8*n - 14)*x + x^2)*P[n - 2, x] - ((4*n - 7)*x + x^2)*D[P[n - 2, x], x] + (8*n - 9 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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