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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":"165da872328444a76bd501ac035ff6110a264c44ad1c3c39b7b4bc187d002cc8","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-ec4e74470f457c51","role":"sequence","name":"shift_x_plus_one of Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.","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:       5,       2;\n  3:      25,      18,       3;\n  4:     133,     134,      42,       4;\n  5:     761,     970,     435,      80,      5;\n  6:    4677,    7142,    4110,    1100,    135,     6;\n  7:   30745,   54362,   37863,   13160,   2380,   210,    7;\n  8:  215173,  430390,  349818,  148484,  35210,  4620,  308,   8;\n  9: 1596217, 3551178, 3285579, 1639008, 478275, 82908, 8274, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4+4*x+x^2)*P(n-2,x) - (4+4*x+x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + (2+x)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(4 + 4*x + x^2)*P[n - 2, x] - (4 + 4*x + x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + (2 + x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 4*x + x^2)*v[i - 2] - (4 + 4*x + x^2)*deriv(v[i - 2], x) + (5 + 2*x)*v[i - 1] + (2 + x)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c15181c2ebe13c1d","steps":["x \u21a6 x+1","D"],"edges":["e-660231fdcb3183b6","e-f634d8cbcbfe2915"],"text":"A049020 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A049020 \u2192 x \u21a6 x+1 \u2192 D","chain":null}