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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":"dd8f3622aab532f59fe24293fb8c8e9f97545d2b13a02b2ecb1dd292efddf357","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-8c29b5d8d22f0df0","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.","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:   1;\n  2:   8,     2;\n  3:  19,    38,      3;\n  4:  35,   316,    105,       4;\n  5:  57,  1184,   1776,     228,       5;\n  6:  86,  3258,  16824,    6516,     430,       6;\n  7: 123,  7534,  85647,  114196,   18835,     738,      7;\n  8: 169, 15520, 316863, 1237832,  528105,   46560,   1183,    8;\n  9: 225, 29388, 961698, 7941096, 9926370, 1923396, 102858, 1800, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+4*n+8)*P(n,x) = -(n^6+3*n^5+3*n^4-3*n^3-4*n^2)*x^2*P(n-4,x) - ((4*n^4+16*n^3+28*n^2+16*n)*x+(4*n^4+12*n^3+20*n^2+12*n)*x^2)*P(n-3,x) - (4*n^2+12*n+16-(4*n^4+16*n^3+20*n^2+8*n-16)*x+(4*n^2+4*n+8)*x^2)*P(n-2,x) + (8*n^2+16*n+24+(8*n^2+8*n+16)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 19 + 38*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 8 + 2*x;\nP[3, x_] := 19 + 38*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*P[n - 4, x] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*P[n - 3, x] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*P[n - 2, x] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*P[n - 1, x])/(4*n^2 + 4*n + 8)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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