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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":"378e3bd4ca1a8019aa404e49313c8c6aa03fb957717c7837eca173b11dc41d3e","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-08fe62d9c34c34e9","role":"sequence","name":"basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:       6;\n  2:      48,       54;\n  3:     288,      720,       324;\n  4:    1536,     6048,      6480,      1620;\n  5:    7680,    41472,     72576,     45360,      7290;\n  6:   36864,   253440,    622080,    653184,    272160,     30618;\n  7:  172032,  1437696,   4561920,   6842880,   4898880,   1469664,    122472;\n  8:  786432,  7741440,  30191616,  59304960,  61585920,  32332608,   7348320,   472392;\n  9: 3538944, 40108032, 185794560, 452874240, 622702080, 480370176, 193995648, 34642080, 1771470;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n+1)*P(n,x) = (8*n^2-4*n+(18*n^2-21*n-3)*x)*P(n-1,x) - (4*n*x+12*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 4*n + (18*n^2 - 21*n - 3)*x)*P[n - 1, x] - (4*n*x + 12*n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 4*n + (18*n^2 - 21*n - 3)*x)*v[i - 1] - (4*n*x + 12*n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 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-2245b35ec7cd4857","steps":["basis A013610","D"],"edges":["e-0cc36dd83eba3acf","e-ac468e34f96f8da4"],"text":"A091044 \u2192 basis A013610 \u2192 D"},"headline":"A091044 \u2192 basis A013610 \u2192 D","chain":null}