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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":"04f850021a12583155c2e0a871729d56b0f6d5fa02168b9e824978e830fc39e4","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-51a2369b25b8e3ba","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).","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:   1;\n  2:   4,    2;\n  3:  10,   12,    3;\n  4:  22,   42,   24,    4;\n  5:  45,  118,  108,   40,    5;\n  6:  88,  294,  372,  220,   60,    6;\n  7: 167,  678, 1098,  900,  390,   84,   7;\n  8: 310, 1482, 2928, 3080, 1850,  630, 112,   8;\n  9: 566, 3114, 7266, 9348, 7215, 3402, 952, 144, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^3-4*n^2-n+4-(2*n^2-2)*x)*P(n,x) = -(2*n^3-22*n^2+58*n-20-(8*n^2-36*n+4)*x+6*n*x^2)*P(n-1,x) + (5*n^2-25*n+10+(6*n^2-40*n+12)*x+(n^2-17*n+2)*x^2-2*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 22*n^2 + 58*n - 20 - (8*n^2 - 36*n + 4)*x + 6*n*x^2)*P[n - 1, x] + (5*n^2 - 25*n + 10 + (6*n^2 - 40*n + 12)*x + (n^2 - 17*n + 2)*x^2 - 2*n*x^3)*D[P[n - 1, x], x])/(n^3 - 4*n^2 - n + 4 - (2*n^2 - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 22*n^2 + 58*n - 20 - (8*n^2 - 36*n + 4)*x + 6*n*x^2)*v[i - 1] + (5*n^2 - 25*n + 10 + (6*n^2 - 40*n + 12)*x + (n^2 - 17*n + 2)*x^2 - 2*n*x^3)*deriv(v[i - 1], x))/(n^3 - 4*n^2 - n + 4 - (2*n^2 - 2)*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-51a2369b25b8e3ba","steps":["D"],"edges":["e-eb56dcacfc1cac25"],"text":"A154929 \u2192 D"},"headline":"A154929 \u2192 D","chain":null}