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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":"fa2b498a9836aa3bcec843cf9f54f19702d0d9f8705a16f4cb783a48b1ef20ec","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-ff69dadb0811f5e6","role":"sequence","name":"Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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:       7,       2;\n  3:      38,      24,       3;\n  4:     192,     202,      54,       4;\n  5:     969,    1510,     645,     100,      5;\n  6:    5115,   10988,    6615,    1600,    165,      6;\n  7:   29322,   81942,   64407,   21420,   3395,    252,     7;\n  8:  187992,  647316,  628068,  266476,  57260,   6468,   364,   8;\n  9: 1370745, 5529852, 6333954, 3254580, 881475, 133812, 11382, 504, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-9*n-6+(6*n-12)*x)*P(n-3,x) - (n^2+8*n-8+(2*n+2)*x+x^2)*P(n-2,x) + (2*n+3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 7 + 2*x, P(3,x) = 38 + 24*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 7 + 2*x;\nP[3, x_] := 38 + 24*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 9*n - 6 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*P[n - 2, x] + (2*n + 3 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 7 + 2*x; v[4] = 38 + 24*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 9*n - 6 + (6*n - 12)*x)*v[i - 3] - (n^2 + 8*n - 8 + (2*n + 2)*x + x^2)*v[i - 2] + (2*n + 3 + 2*x)*v[i - 1]); 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-ff69dadb0811f5e6","steps":["D"],"edges":["e-900b162596e739b8"],"text":"A327997 \u2192 D"},"headline":"A327997 \u2192 D","chain":null}