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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":"2dec7001748c97f070cfdbc36e642a8817b9e28e869bc8d79fb4e97810b39bd1","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-e0a0f9191d8a0512","role":"sequence","name":"basis_map_A007318 of 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:        9,        2;\n  3:       65,       30,        3;\n  4:      452,      322,       66,       4;\n  5:     3229,     3120,      975,     120,       5;\n  6:    24489,    29708,    12465,    2320,     195,      6;\n  7:   200745,   289898,   151662,   37660,    4760,    294,     7;\n  8:  1793952,  2966500,  1841364,  567756,   95340,   8820,   420,   8;\n  9: 17516313, 32228352, 22906230, 8364384, 1739535, 213192, 15162, 576, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-3*n-18+(6*n-12)*x)*P(n-3,x) - (n^2+10*n-5+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 9 + 2*x, P(3,x) = 65 + 30*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 9 + 2*x;\nP[3, x_] := 65 + 30*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 3*n - 18 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 10*n - 5 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 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] = 9 + 2*x; v[4] = 65 + 30*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 - 3*n - 18 + (6*n - 12)*x)*v[i - 3] - (n^2 + 10*n - 5 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 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":["x \u21a6 x+1","D"],"edges":["e-29bcfb0ecbd3591f","e-209065cba64dbf45"],"text":"A327997 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A327997 \u2192 x \u21a6 x+1 \u2192 D","chain":null}