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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":"7f5a2a8db94e1e707899b3cc02de42db82be864240e23c7a257400a743b9a379","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e9164c616dcc1bbd","role":"sequence","name":"Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        0;\n   2:        1;\n   3:        6,         2;\n   4:       38,        30,         3;\n   5:      272,       376,        84,        4;\n   6:     2200,      4680,      1740,      180,        5;\n   7:    19920,     60560,     33120,     5560,      330,       6;\n   8:   199920,    827120,    620760,   149800,    14210,     546,     7;\n   9:  2204160,  11979520,  11793600,  3823680,   514080,   31248,   840,    8;\n  10: 26490240, 184101120, 230254080, 96022080, 17117520, 1460592, 61656, 1224, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-n-2+(2*n-4)*x+x^2)*P(n-2,x) - ((3*n-6)*x+3*x^2)*d/dx P(n-2,x) + (2*n+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - n - 2 + (2*n - 4)*x + x^2)*P[n - 2, x] - ((3*n - 6)*x + 3*x^2)*D[P[n - 2, x], x] + (2*n + 2*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(n^2 - n - 2 + (2*n - 4)*x + x^2)*v[i - 2] - ((3*n - 6)*x + 3*x^2)*deriv(v[i - 2], x) + (2*n + 2*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e9164c616dcc1bbd","steps":["D"],"edges":["e-fd558332af976d79"],"text":"A322944 \u2192 D"},"headline":"A322944 \u2192 D","chain":null}