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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":"33ef2c46d809b02b3e25d73f8b109b5423469df5037d1429f299c2ebefff5ee5","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-c47b645304a2946e","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.","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:        4;\n  2:       40,        32;\n  3:      296,       592,        192;\n  4:     1928,      7104,       5784,       1024;\n  5:    11688,     69616,     104424,      46752,       5120;\n  6:    67656,    604704,    1461936,    1209408,     338280,      24576;\n  7:   379240,   4846032,   17490000,   23320000,   12115080,    2275440,    114688;\n  8:  2076424,  36660992,  187851432,  373120000,  313085720,  109982976,  14534968,   524288;\n  9: 11168552, 265647280, 1864874904, 5239177184, 6548971480, 3729749808, 929765480, 89348416, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(20*n+60+(8*n^2+32*n+24)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((4*n+12)*x+(n^2+n-6)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (9*n+22+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(20*n + 60 + (8*n^2 + 32*n + 24)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((4*n + 12)*x + (n^2 + n - 6)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (9*n + 22 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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