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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":"003a58a1d79a0130a89b735c260f5ba77f3d732ba8d8f2bead3acf7b5f642627","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7697b772cd7a5e3f","role":"sequence","name":"theta_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 2,  2;\n   3: 2,  8,  3;\n   4: 0, 12, 18,   4;\n   5: 0,  6, 39,  32,   5;\n   6: 0,  0, 36,  92,  50,   6;\n   7: 0,  0, 12, 124, 180,  72,    7;\n   8: 0,  0,  0,  80, 320, 312,   98,   8;\n   9: 0,  0,  0,  20, 305, 690,  497, 128,   9;\n  10: 0,  0,  0,   0, 150, 870, 1316, 744, 162, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^3-4*n^2+5*n-2+(2*n^2-8*n+6)*x+(n^2-2*n+1)*x^2+(n-1)*x^3)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(8*n-20)*x+(2*n^2-8*n+10)*x^2-4*x^3-2*x^4)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2+5*n-14)*x^2+6*n*x^3+(n+5)*x^4+x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n - 20)*x + (2*n^2 - 8*n + 10)*x^2 - 4*x^3 - 2*x^4)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 + 5*n - 14)*x^2 + 6*n*x^3 + (n + 5)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 5*n - 2 + (2*n^2 - 8*n + 6)*x + (n^2 - 2*n + 1)*x^2 + (n - 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n - 20)*x + (2*n^2 - 8*n + 10)*x^2 - 4*x^3 - 2*x^4)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 + 5*n - 14)*x^2 + 6*n*x^3 + (n + 5)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 5*n - 2 + (2*n^2 - 8*n + 6)*x + (n^2 - 2*n + 1)*x^2 + (n - 1)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-392f49b496f79fbf","steps":["\u03b8+1","D+1"],"edges":["e-cd00da36b6a89aa7","e-9003ccac11c5cb61"],"text":"A026729 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A026729 \u2192 \u03b8+1 \u2192 D+1","chain":null}