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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":"e0044d352291b37f76977b975eb835e714b09b425179a205c935effcfb5d9836","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-daa3b3e1ead00edc","role":"sequence","name":"Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.","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:      13,         2;\n   4:      93,       120,          3;\n   5:     545,      2262,        753,           4;\n   6:    2933,     28996,      32034,        4072,           5;\n   7:   15177,    308226,     787314,      355992,       20445,          6;\n   8:   77101,   2951472,   14670861,    15481408,     3462495,      98256,         7;\n   9:  388321,  26541614,  232214799,   471034636,   252380015,   31050078,    458689,       8;\n  10: 1948005, 229419036, 3310798788, 11548400616, 11944773090, 3649561740, 264125484, 2097072, 9;\n","recurrence_sentence":"The row polynomials satisfy (3*n-8)*P(n,x) = -(30*n-50+(36*n^2-192*n+220)*x+(27*n^3-216*n^2+549*n-440)*x^2)*P(n-2,x) - ((18*n-30)*x+(27*n^2-135*n+150)*x^2-(27*n^2-117*n+120)*x^3)*d/dx P(n-2,x) + (21*n-50+(18*n^2-96*n+128)*x)*P(n-1,x) + ((9*n-24)*x-(9*n-24)*x^2)*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[Together[(-(30*n - 50 + (36*n^2 - 192*n + 220)*x + (27*n^3 - 216*n^2 + 549*n - 440)*x^2)*P[n - 2, x] - ((18*n - 30)*x + (27*n^2 - 135*n + 150)*x^2 - (27*n^2 - 117*n + 120)*x^3)*D[P[n - 2, x], x] + (21*n - 50 + (18*n^2 - 96*n + 128)*x)*P[n - 1, x] + ((9*n - 24)*x - (9*n - 24)*x^2)*D[P[n - 1, x], x])/(3*n - 8)]];\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] = (-(30*n - 50 + (36*n^2 - 192*n + 220)*x + (27*n^3 - 216*n^2 + 549*n - 440)*x^2)*v[i - 2] - ((18*n - 30)*x + (27*n^2 - 135*n + 150)*x^2 - (27*n^2 - 117*n + 120)*x^3)*deriv(v[i - 2], x) + (21*n - 50 + (18*n^2 - 96*n + 128)*x)*v[i - 1] + ((9*n - 24)*x - (9*n - 24)*x^2)*deriv(v[i - 1], x))/(3*n - 8)); 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-daa3b3e1ead00edc","steps":["D"],"edges":["e-af6e8a04536e52d2"],"text":"A225117 \u2192 D"},"headline":"A225117 \u2192 D","chain":null}