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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":"c52d5ceccce9c5b2e10b80535820e8dbc7d97d3304212511772e97c9108d70b3","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-ff3909e492eb580d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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:     1;\n  1:     8,      3;\n  2:    39,     33,      6;\n  3:   152,    204,     84,     10;\n  4:   520,    960,    630,    170,     15;\n  5:  1632,   3840,   3480,   1500,    300,     21;\n  6:  4816,  13776,  15960,   9520,   3045,    483,    28;\n  7: 13568,  45696,  64512,  49280,  21840,   5544,   728,    36;\n  8: 36864, 142848, 237888, 221760, 126000,  44352,  9324,  1044,   45;\n  9: 97280, 426240, 817920, 900480, 624960, 282240, 82320, 14760, 1440, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16+32*x+24*x^2+8*x^3+x^4)*P(n-4,x) + (32+48*x+24*x^2+4*x^3)*P(n-3,x) - (24+24*x+6*x^2)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 8 + 3*x, P(2,x) = 39 + 33*x + 6*x^2, P(3,x) = 152 + 204*x + 84*x^2 + 10*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + 3*x;\nP[2, x_] := 39 + 33*x + 6*x^2;\nP[3, x_] := 152 + 204*x + 84*x^2 + 10*x^3;\nP[n_, x_] := P[n, x] = Expand[-(16 + 32*x + 24*x^2 + 8*x^3 + x^4)*P[n - 4, x] + (32 + 48*x + 24*x^2 + 4*x^3)*P[n - 3, x] - (24 + 24*x + 6*x^2)*P[n - 2, x] + (8 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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