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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":"b8d605df953830862b877e41f5ec035c214cb158f2499be354ec1bd13f7d764f","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-551f44a89cf7b3e1","role":"sequence","name":"basis_map_A013610 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:        9,         3;\n  2:       71,        54,         9;\n  3:      522,       642,       243,        27;\n  4:     3673,      6318,      3861,       972,        81;\n  5:    25083,     55737,     47628,     19332,      3645,       243;\n  6:   167615,    457812,    505494,    286740,     87075,     13122,      729;\n  7:  1101780,   3575652,   4854654,   3557790,   1509030,    365958,    45927,    2187;\n  8:  7149313,  26900532,  43413318,  39123972,  21433815,   7256466,  1464561,  157464,   6561;\n  9: 45910413, 196607103, 368011836, 394277652, 265595841, 116108559, 32700024, 5651208, 531441, 19683;\n","recurrence_sentence":"The row polynomials satisfy (54*n^3+78*n-54*n*x-54*n*x^2)*P(n,x) = (162*n^3+486*n^2+432*n+108+(81*n^3+243*n^2-126*n-288)*x-(324*n+324)*x^2-(81*n+81)*x^3)*P(n-1,x) + (288*n^2+288*n-448+(324*n^2+324*n-1368)*x+(81*n^2+81*n-1386)*x^2-567*x^3-81*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[((162*n^3 + 486*n^2 + 432*n + 108 + (81*n^3 + 243*n^2 - 126*n - 288)*x - (324*n + 324)*x^2 - (81*n + 81)*x^3)*P[n - 1, x] + (288*n^2 + 288*n - 448 + (324*n^2 + 324*n - 1368)*x + (81*n^2 + 81*n - 1386)*x^2 - 567*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 + 78*n - 54*n*x - 54*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] = ((162*n^3 + 486*n^2 + 432*n + 108 + (81*n^3 + 243*n^2 - 126*n - 288)*x - (324*n + 324)*x^2 - (81*n + 81)*x^3)*v[i - 1] + (288*n^2 + 288*n - 448 + (324*n^2 + 324*n - 1368)*x + (81*n^2 + 81*n - 1386)*x^2 - 567*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 + 78*n - 54*n*x - 54*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 A013610","D+1"],"edges":["e-d13df69ca10128cd","e-ffe03a7ffb0946e2"],"text":"A207824 \u2192 basis A013610 \u2192 D+1"},"headline":"A207824 \u2192 basis A013610 \u2192 D+1","chain":null}