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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":"c7327eaa9c89035f38433de1b8cd15e11cf1c26a717b767549f79ac95f982cff","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-c2b7c74f8491f01b","role":"sequence","name":"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:       6,       1;\n  2:      34,      12,       1;\n  3:     188,     103,      18,       1;\n  4:    1021,     764,     207,      24,      1;\n  5:    5468,    5208,    1928,     346,     30,      1;\n  6:   28959,   33572,   15831,    3880,    520,     36,     1;\n  7:  151976,  207921,  119430,   37285,   6820,    729,    42,    1;\n  8:  791526, 1249380,  847515,  322360,  75090,  10948,   973,   48,  1;\n  9: 4096030, 7331655, 5741640, 2579830, 732130, 135891, 16464, 1252, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-14*n-14*n*x-2*n*x^2)*P(n,x) = (5*n^3+15*n^2-41*n-51+(n^3+3*n^2-49*n-51)*x-(13*n+13)*x^2-(n+1)*x^3)*P(n-1,x) + (21*n^2+21*n-336+(10*n^2+10*n-349)*x+(n^2+n-127)*x^2-19*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[((5*n^3 + 15*n^2 - 41*n - 51 + (n^3 + 3*n^2 - 49*n - 51)*x - (13*n + 13)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (21*n^2 + 21*n - 336 + (10*n^2 + 10*n - 349)*x + (n^2 + n - 127)*x^2 - 19*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 14*n - 14*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] = ((5*n^3 + 15*n^2 - 41*n - 51 + (n^3 + 3*n^2 - 49*n - 51)*x - (13*n + 13)*x^2 - (n + 1)*x^3)*v[i - 1] + (21*n^2 + 21*n - 336 + (10*n^2 + 10*n - 349)*x + (n^2 + n - 127)*x^2 - 19*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 14*n - 14*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":["D+1"],"edges":["e-876ed73f7b487e82"],"text":"A207824 \u2192 D+1"},"headline":"A207824 \u2192 D+1","chain":null}