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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":"feab69c1fa47d7084818fb0459423bb7d79c9466d0d8a135ebce6138d36cee27","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-677d34c7af372519","role":"sequence","name":"Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, 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:   1;\n  1:   2,     1;\n  2:   4,     7,      2;\n  3:   8,    31,     30,       6;\n  4:  16,   115,    230,     156,       24;\n  5:  32,   391,   1350,    1830,      960,      120;\n  6:  64,  1267,   6902,   15540,    15960,     6840,      720;\n  7: 128,  3991,  32550,  111006,   184800,   152880,    55440,     5040;\n  8: 256, 12355, 145910,  713916,  1754424,  2313360,  1602720,   504000,   40320;\n  9: 512, 37831, 632550, 4287030, 14716800, 28140840, 30693600, 18295200, 5080320, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+3*x-x^3)*P(n-2,x) - (x+x^2-x^3-x^4)*d/dx P(n-2,x) + (3+3*x-x^2)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(2 + 3*x - x^3)*P[n - 2, x] - (x + x^2 - x^3 - x^4)*D[P[n - 2, x], x] + (3 + 3*x - x^2)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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