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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":"a854050d5a456cdbf8bc2eff5c48e0afadf2c7568cc80b0fa8a6b3e91114ab41","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-359eaa7b22014e19","role":"sequence","name":"basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:      3;\n  2:     18,      12;\n  3:     87,     114,      36;\n  4:    378,     738,     468,      96;\n  5:   1548,    4014,    3825,    1584,     240;\n  6:   6102,   19728,   25092,   15672,    4800,     576;\n  7:  23421,   90702,  144306,  120612,   55800,   13536,   1344;\n  8:  88146,  397722,  759708,  795780,  493320,  180864,  36288,  3072;\n  9: 326769, 1683294, 3752649, 4726248, 3675975, 1807056, 548016, 93696, 6912;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(2*n-2)*x)*P(n,x) = -(5*n-16+(4*n-24)*x-8*x^2)*P(n-1,x) + (13+29*x+20*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(5*n - 16 + (4*n - 24)*x - 8*x^2)*P[n - 1, x] + (13 + 29*x + 20*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(5*n - 16 + (4*n - 24)*x - 8*x^2)*v[i - 1] + (13 + 29*x + 20*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x)); 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-e7331b0d50084c15","steps":["basis A007318","D"],"edges":["e-212843e2c5c5a487","e-53e552fae5d8c928"],"text":"A249251 \u2192 basis A007318 \u2192 D"},"headline":"A249251 \u2192 basis A007318 \u2192 D","chain":null}