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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":"54ead70a32aff9c39a4636e6fe38d87e6a5c6626a2e59e2223568d4742cdd5d6","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-cca535f6115f5fc0","role":"sequence","name":"basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/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:       1;\n  1:       5,       2;\n  2:      26,      20,       3;\n  3:     128,     154,      49,       4;\n  4:     608,    1008,     524,      96,       5;\n  5:    2816,    5984,    4416,    1356,     165,       6;\n  6:   12800,   33280,   32128,   14288,    2966,     260,      7;\n  7:   57344,  176640,  211712,  125056,   38144,    5782,    385,     8;\n  8:  253952,  905216, 1299456,  964096,  395200,   88992,  10360,   544,   9;\n  9: 1114112, 4513792, 7561216, 6779904, 3523840, 1076416, 187776, 17400, 741, 10;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-7*n^2-11*n+12-(2*n^3-11*n^2+24*n-15)*x-(4*n^2-9*n+6)*x^2-2*n*x^3)*P(n,x) = -(24*n^3-4*n^2-104*n+84+(10*n^3+29*n^2-143*n+104)*x-(6*n^3-15*n^2+12*n+2)*x^2-(12*n^2-17*n+17)*x^3-(6*n-2)*x^4)*P(n-1,x) + ((12*n^2+16*n-28)*x+(8*n^2+32*n-30)*x^2-(4*n^2-8*n-8)*x^3-(8*n-6)*x^4-4*x^5)*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[(-(24*n^3 - 4*n^2 - 104*n + 84 + (10*n^3 + 29*n^2 - 143*n + 104)*x - (6*n^3 - 15*n^2 + 12*n + 2)*x^2 - (12*n^2 - 17*n + 17)*x^3 - (6*n - 2)*x^4)*P[n - 1, x] + ((12*n^2 + 16*n - 28)*x + (8*n^2 + 32*n - 30)*x^2 - (4*n^2 - 8*n - 8)*x^3 - (8*n - 6)*x^4 - 4*x^5)*D[P[n - 1, x], x])/(-6*n^3 + 7*n^2 + 11*n - 12 + (2*n^3 - 11*n^2 + 24*n - 15)*x + (4*n^2 - 9*n + 6)*x^2 + 2*n*x^3)]];\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] = (-(24*n^3 - 4*n^2 - 104*n + 84 + (10*n^3 + 29*n^2 - 143*n + 104)*x - (6*n^3 - 15*n^2 + 12*n + 2)*x^2 - (12*n^2 - 17*n + 17)*x^3 - (6*n - 2)*x^4)*v[i - 1] + ((12*n^2 + 16*n - 28)*x + (8*n^2 + 32*n - 30)*x^2 - (4*n^2 - 8*n - 8)*x^3 - (8*n - 6)*x^4 - 4*x^5)*deriv(v[i - 1], x))/(-6*n^3 + 7*n^2 + 11*n - 12 + (2*n^3 - 11*n^2 + 24*n - 15)*x + (4*n^2 - 9*n + 6)*x^2 + 2*n*x^3)); 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-c85abc56078671cc","steps":["basis A007318","D+1"],"edges":["e-2b8d1605acbc4584","e-92f9ed71d54a49fc"],"text":"A232535 \u2192 basis A007318 \u2192 D+1"},"headline":"A232535 \u2192 basis A007318 \u2192 D+1","chain":null}