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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":"70d0890c25352bc686662bb3a97c0255b94202131b492c43ce95aba783108e96","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-1fb3d5de4e4b154d","role":"sequence","name":"basis_map_A007318 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:        7,        1;\n  2:       47,       14,        1;\n  3:      310,      142,       21,       1;\n  4:     2017,     1254,      285,      28,       1;\n  5:    12981,    10227,     3156,     476,      35,      1;\n  6:    82799,    79140,    30966,    6340,     715,     42,     1;\n  7:   524204,   589820,   280146,   72730,   11130,   1002,    49,    1;\n  8:  3297841,  4272772,  2390246,  753396,  146175,  17850,  1337,   56,  1;\n  9: 20634947, 30270389, 19507620, 7243468, 1706271, 264117, 26824, 1720, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-30*n-18*n*x-2*n*x^2)*P(n,x) = (6*n^3+18*n^2-104*n-116+(n^3+3*n^2-78*n-80)*x-(16*n+16)*x^2-(n+1)*x^3)*P(n-1,x) + (32*n^2+32*n-832+(12*n^2+12*n-664)*x+(n^2+n-190)*x^2-23*x^3-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 - 104*n - 116 + (n^3 + 3*n^2 - 78*n - 80)*x - (16*n + 16)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (32*n^2 + 32*n - 832 + (12*n^2 + 12*n - 664)*x + (n^2 + n - 190)*x^2 - 23*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 30*n - 18*n*x - 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 - 104*n - 116 + (n^3 + 3*n^2 - 78*n - 80)*x - (16*n + 16)*x^2 - (n + 1)*x^3)*v[i - 1] + (32*n^2 + 32*n - 832 + (12*n^2 + 12*n - 664)*x + (n^2 + n - 190)*x^2 - 23*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 30*n - 18*n*x - 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":["x \u21a6 x+1","D+1"],"edges":["e-1b36e9a96cf70567","e-22976fb6cb3c5fc9"],"text":"A207824 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A207824 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}