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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":"bb86787192fb5b1cd80bea233cb5780b4296571805b803f84421215bfe27baf1","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-cc665cf2fd1f1fd7","role":"sequence","name":"gamma_polynomial of Pascal-(1,4,1) array.","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:  1;\n  1:  1;\n  2:  5,   4;\n  3:  9,   8;\n  4: 13,  44,   16;\n  5: 17, 112,   48;\n  6: 21, 212,  288,   64;\n  7: 25, 344,  928,  256;\n  8: 29, 508, 2160, 1664,  256;\n  9: 33, 704, 4176, 6400, 1280;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-11*n^2+7*n+(16*n^2-39*n)*x+16*n*x^2)*P(n,x) = (4*n^3-7*n^2+3*n+(32*n^3-104*n^2+49*n)*x+(128*n^2-296*n+184)*x^2+(128*n-128)*x^3)*P(n-1,x) - ((4*n^2-3*n)*x+(64*n^2-32*n-7)*x^2+(256*n-96)*x^3+256*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 7*n^2 + 3*n + (32*n^3 - 104*n^2 + 49*n)*x + (128*n^2 - 296*n + 184)*x^2 + (128*n - 128)*x^3)*P[n - 1, x] - ((4*n^2 - 3*n)*x + (64*n^2 - 32*n - 7)*x^2 + (256*n - 96)*x^3 + 256*x^4)*D[P[n - 1, x], x])/(4*n^3 - 11*n^2 + 7*n + (16*n^2 - 39*n)*x + 16*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 - 7*n^2 + 3*n + (32*n^3 - 104*n^2 + 49*n)*x + (128*n^2 - 296*n + 184)*x^2 + (128*n - 128)*x^3)*v[i - 1] - ((4*n^2 - 3*n)*x + (64*n^2 - 32*n - 7)*x^2 + (256*n - 96)*x^3 + 256*x^4)*deriv(v[i - 1], x))/(4*n^3 - 11*n^2 + 7*n + (16*n^2 - 39*n)*x + 16*n*x^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-ed9434bfd889c409","steps":["\u03b3","D+1"],"edges":["e-5cd8190965f3efc9","e-02a4e034b685f48c"],"text":"A081579 \u2192 \u03b3 \u2192 D+1"},"headline":"A081579 \u2192 \u03b3 \u2192 D+1","chain":null}