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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":"1172cc92101ebfd50324026587e58dde2bc45435f4daf0c35e1893ca5a4f7f8e","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-f1ff6363886186a4","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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:          2;\n  2:         14,           8;\n  3:        132,         120,         24;\n  4:       1612,        1848,        624,         64;\n  5:      24328,       32320,      14280,       2560,       160;\n  6:     438216,      646752,     339720,      81600,      9120,      384;\n  7:    9183200,    14687904,    8734656,    2537920,    386400,    29568,     896;\n  8:  219591200,   374554592,  244960800,   81664576,  15232000,  1607424,   89600,   2048;\n  9: 5902056000, 10616784192, 7492644576, 2776507776, 600193440, 78349824, 6080256, 258048, 4608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(12*n-16)*x+4*x^2)*P(n-2,x) + (6*n-5+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (12*n - 16)*x + 4*x^2)*P[n - 2, x] + (6*n - 5 + 4*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (12*n - 16)*x + 4*x^2)*v[i - 2] + (6*n - 5 + 4*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-260e022530e5976c","steps":["basis A013609","D"],"edges":["e-dabb8392a3e47d84","e-8d1525ee334721af"],"text":"A286718 \u2192 basis A013609 \u2192 D"},"headline":"A286718 \u2192 basis A013609 \u2192 D","chain":null}