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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":"fe0c9d15b62bfbb8e4a6b6c5b0f1b51259786c3733c72bd204aa87e74fba7ddf","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-b5ebaf3676d81658","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).","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:    0;\n  2:    3;\n  3:    6;\n  4:   21,   18;\n  5:   48,   54;\n  6:  123,  216,   81;\n  7:  282,  612,  324;\n  8:  657, 1782, 1458,  324;\n  9: 1476, 4698, 4860, 1620;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (5*n-13+(6*n-18)*x)*P(n-1,x) - (6+17*x+12*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n - 13 + (6*n - 18)*x)*P[n - 1, x] - (6 + 17*x + 12*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 3; for(n = 3, N, my(i = n + 1); v[i] = ((5*n - 13 + (6*n - 18)*x)*v[i - 1] - (6 + 17*x + 12*x^2)*deriv(v[i - 1], x))/(n - 2)); 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-7d332fcb7bf9a0f7","steps":["basis A013610","D"],"edges":["e-95533c5935436804","e-b9a61af70b276272"],"text":"A128100 \u2192 basis A013610 \u2192 D"},"headline":"A128100 \u2192 basis A013610 \u2192 D","chain":null}