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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":"0507330c9d306cf77aa23b65bd895d084063866aac7f0fa7527570ea8602abe3","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-c85abc56078671cc","role":"sequence","name":"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:   0;\n  1:   2;\n  2:   8,    6;\n  3:  18,   50,    12;\n  4:  32,  196,   168,     20;\n  5:  50,  540,  1008,    420,     30;\n  6:  72, 1210,  3960,   3564,    880,     42;\n  7:  98, 2366, 12012,  18876,  10010,   1638,    56;\n  8: 128, 4200, 30576,  74360,  68640,  24024,  2800,   72;\n  9: 162, 6936, 68544, 238680, 340340, 206856, 51408, 4488, 90;\n","recurrence_sentence":"The row polynomials satisfy -(8*n^3-22*n^2+20*n-6-(2*n^3-3*n^2+n)*x)*P(n,x) = -(8*n^3-6*n^2+(22*n^3-53*n^2+15*n+12)*x-(6*n^3-7*n^2-n)*x^2)*P(n-1,x) - ((16*n^2-8*n-4)*x-(20*n^2-8*n-4)*x^2+4*n^2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 6*n^2 + (22*n^3 - 53*n^2 + 15*n + 12)*x - (6*n^3 - 7*n^2 - n)*x^2)*P[n - 1, x] - ((16*n^2 - 8*n - 4)*x - (20*n^2 - 8*n - 4)*x^2 + 4*n^2*x^3)*D[P[n - 1, x], x])/(-8*n^3 + 22*n^2 - 20*n + 6 + (2*n^3 - 3*n^2 + n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 6*n^2 + (22*n^3 - 53*n^2 + 15*n + 12)*x - (6*n^3 - 7*n^2 - n)*x^2)*v[i - 1] - ((16*n^2 - 8*n - 4)*x - (20*n^2 - 8*n - 4)*x^2 + 4*n^2*x^3)*deriv(v[i - 1], x))/(-8*n^3 + 22*n^2 - 20*n + 6 + (2*n^3 - 3*n^2 + n)*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-c85abc56078671cc","steps":["D"],"edges":["e-16137bf2f4ea840e"],"text":"A232535 \u2192 D"},"headline":"A232535 \u2192 D","chain":null}