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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":"42d22d0f083e8748688730ac543878edad269134a66d7498309f1ad0e2dce796","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c1716beb5dbd6d37","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      3,       1;\n   3:     16,      10,       1;\n   4:     80,      84,      21,       1;\n   5:    384,     576,     256,      36,       1;\n   6:   1792,    3520,    2320,     600,      55,      1;\n   7:   8192,   19968,   17664,    6944,    1200,     78,     1;\n   8:  36864,  107520,  120064,   64512,   17248,   2156,   105,    1;\n   9: 163840,  557056,  753664,  518144,  192000,  37632,  3584,  136,   1;\n  10: 720896, 2801664, 4460544, 3753984, 1799424, 494208, 74592, 5616, 171, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-9*n^2+10*n+(4*n^2-17*n+18)*x+(2*n-4)*x^2)*P(n,x) = (8*n^3-28*n^2+4*n+40+(6*n^3-7*n^2-53*n+86)*x+(12*n^2-37*n+26)*x^2+(6*n-12)*x^3)*P(n-1,x) - ((4*n^2-4*n-8)*x+(4*n^2+4*n-18)*x^2+(8*n-6)*x^3+4*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 28*n^2 + 4*n + 40 + (6*n^3 - 7*n^2 - 53*n + 86)*x + (12*n^2 - 37*n + 26)*x^2 + (6*n - 12)*x^3)*P[n - 1, x] - ((4*n^2 - 4*n - 8)*x + (4*n^2 + 4*n - 18)*x^2 + (8*n - 6)*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 9*n^2 + 10*n + (4*n^2 - 17*n + 18)*x + (2*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = ((8*n^3 - 28*n^2 + 4*n + 40 + (6*n^3 - 7*n^2 - 53*n + 86)*x + (12*n^2 - 37*n + 26)*x^2 + (6*n - 12)*x^3)*v[i - 1] - ((4*n^2 - 4*n - 8)*x + (4*n^2 + 4*n - 18)*x^2 + (8*n - 6)*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 9*n^2 + 10*n + (4*n^2 - 17*n + 18)*x + (2*n - 4)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A007318","D+1"],"edges":["e-219ad3693217b36c","e-7231c8398fee2e9e"],"text":"A086645 \u2192 basis A007318 \u2192 D+1"},"headline":"A086645 \u2192 basis A007318 \u2192 D+1","chain":null}