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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":"8666c17dcc644dff4afa722d846ba096321970d05d50c07f6c790440c7986377","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-991b9bffcf8ddedb","role":"sequence","name":"Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","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:    2;\n  3:    8;\n  4:   22,    32;\n  5:   52,   272;\n  6:  114,  1440,    816;\n  7:  240,  6144,  11904;\n  8:  494, 23232, 102912,  31744;\n  9: 1004, 81552, 690432, 707584;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(2*n-2+(4*n^2-20*n+16)*x+(4*n^3-28*n^2+56*n-32)*x^2)*P(n-2,x) - ((n-1)*x+(2*n^2-10*n+8)*x^2-(8*n^2-24*n+16)*x^3)*d/dx P(n-2,x) + (3*n-5+(4*n^2-20*n+24)*x)*P(n-1,x) + ((n-2)*x-(4*n-8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (4*n^2 - 20*n + 16)*x + (4*n^3 - 28*n^2 + 56*n - 32)*x^2)*P[n - 2, x] - ((n - 1)*x + (2*n^2 - 10*n + 8)*x^2 - (8*n^2 - 24*n + 16)*x^3)*D[P[n - 2, x], x] + (3*n - 5 + (4*n^2 - 20*n + 24)*x)*P[n - 1, x] + ((n - 2)*x - (4*n - 8)*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] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n - 2 + (4*n^2 - 20*n + 16)*x + (4*n^3 - 28*n^2 + 56*n - 32)*x^2)*v[i - 2] - ((n - 1)*x + (2*n^2 - 10*n + 8)*x^2 - (8*n^2 - 24*n + 16)*x^3)*deriv(v[i - 2], x) + (3*n - 5 + (4*n^2 - 20*n + 24)*x)*v[i - 1] + ((n - 2)*x - (4*n - 8)*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-ff270bcb802a85af","steps":["\u03b3 (shift +1)","D"],"edges":["e-e49ea542bfed6bbf","e-34f81fb2d053d311"],"text":"A008292 \u2192 \u03b3 (shift +1) \u2192 D"},"headline":"A008292 \u2192 \u03b3 (shift +1) \u2192 D","chain":null}