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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":"089c401e3526f943d280cdeef9e038623cbea6ce3810303290eac111c0e0603f","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-a1e0668357b9b581","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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:       10,         4;\n  2:       83,        80,        16;\n  3:      628,      1000,       480,         64;\n  4:     4501,     10128,      8016,       2560,        256;\n  5:    31134,     91020,    101760,      53504,      12800,      1024;\n  6:   210023,    757344,   1100256,     816640,     321280,     61440,      4096;\n  7:  1390568,   5971664,  10704576,   10322560,    5729280,   1800192,    286720,    16384;\n  8:  9075049,  45255520,  96646880,  114998784,   82901760,  36728832,   9605120,  1310720,   65536;\n  9: 58548146, 332673428, 825303936, 1170085120, 1040718336, 598692864, 220659712, 49414144, 5898240, 262144;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+3*n-2*n*x^2)*P(n,x) = (6*n^3+18*n^2+19*n+7+(4*n^3+12*n^2+6*n-2)*x-(10*n+10)*x^2-(4*n+4)*x^3)*P(n-1,x) + (8*n^2+8*n-4+(12*n^2+12*n-22)*x+(4*n^2+4*n-34)*x^2-20*x^3-4*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n^3 + 18*n^2 + 19*n + 7 + (4*n^3 + 12*n^2 + 6*n - 2)*x - (10*n + 10)*x^2 - (4*n + 4)*x^3)*P[n - 1, x] + (8*n^2 + 8*n - 4 + (12*n^2 + 12*n - 22)*x + (4*n^2 + 4*n - 34)*x^2 - 20*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 + 3*n - 2*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6*n^3 + 18*n^2 + 19*n + 7 + (4*n^3 + 12*n^2 + 6*n - 2)*x - (10*n + 10)*x^2 - (4*n + 4)*x^3)*v[i - 1] + (8*n^2 + 8*n - 4 + (12*n^2 + 12*n - 22)*x + (4*n^2 + 4*n - 34)*x^2 - 20*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 + 3*n - 2*n*x^2)); 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-c2b7c74f8491f01b","steps":["basis A013611","D+1"],"edges":["e-3938f5b848156b83","e-dc2ec16d86ba9683"],"text":"A207824 \u2192 basis A013611 \u2192 D+1"},"headline":"A207824 \u2192 basis A013611 \u2192 D+1","chain":null}
