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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":"3073aedecc6de1a35765de76fe0343e2b84b14efb5b7eafd9b1c4156a561a8c7","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:        0;\n  1:        1;\n  2:       12,        2;\n  3:      106,       36,        3;\n  4:      828,      426,       72,       4;\n  5:     6051,     4176,     1068,     120,      5;\n  6:    42408,    36732,    12600,    2140,    180,      6;\n  7:   288788,   301032,   129630,   29520,   3750,    252,    7;\n  8:  1925736,  2347036,  1216728,  347820,  59220,   6006,  336,   8;\n  9: 12637733, 17632656, 10691292, 3679704, 786345, 106848, 9016, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (6*n+12+(n+2)*x)*P(n-1,x) + (32+12*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 12 + (n + 2)*x)*P[n - 1, x] + (32 + 12*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 12 + (n + 2)*x)*v[i - 1] + (32 + 12*x + x^2)*deriv(v[i - 1], x))/(2*n - 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"],"edges":["e-1b36e9a96cf70567","e-9d8f534431bf5998"],"text":"A207824 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A207824 \u2192 x \u21a6 x+1 \u2192 D","chain":null}