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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":"3521e569422d6837b9e4481abe9661aaa4cc67f43b3d1d93a4a42f314fdafcf7","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-0bad942bfffd4f03","role":"sequence","name":"basis_map_A007318 of Pascal-(1,4,1) array.","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:      8,      2;\n  3:     36,     28,      3;\n  4:    160,    200,     60,      4;\n  5:    640,   1120,    600,    104,      5;\n  6:   2496,   5600,   4320,   1344,    160,     6;\n  7:   9408,  25792,  26160,  12064,   2540,   228,    7;\n  8:  34816, 112768, 141888,  87424,  27520,  4296,  308,   8;\n  9: 126720, 473600, 712320, 551808, 234800, 54720, 6720, 400, 9;\n","recurrence_sentence":"The row polynomials satisfy (10*n-10+(5*n^2-5*n)*x-(n-1)*x^2)*P(n,x) = -(50*n^2-210*n+140+(45*n^2-225*n+170)*x-(24*n-36)*x^2+2*x^3)*P(n-1,x) + (100*n-100+(200*n-220)*x+(105*n-145)*x^2+(5*n-26)*x^3-x^4)*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[(-(50*n^2 - 210*n + 140 + (45*n^2 - 225*n + 170)*x - (24*n - 36)*x^2 + 2*x^3)*P[n - 1, x] + (100*n - 100 + (200*n - 220)*x + (105*n - 145)*x^2 + (5*n - 26)*x^3 - x^4)*D[P[n - 1, x], x])/(10*n - 10 + (5*n^2 - 5*n)*x - (n - 1)*x^2)]];\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] = (-(50*n^2 - 210*n + 140 + (45*n^2 - 225*n + 170)*x - (24*n - 36)*x^2 + 2*x^3)*v[i - 1] + (100*n - 100 + (200*n - 220)*x + (105*n - 145)*x^2 + (5*n - 26)*x^3 - x^4)*deriv(v[i - 1], x))/(10*n - 10 + (5*n^2 - 5*n)*x - (n - 1)*x^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-ed9434bfd889c409","steps":["x \u21a6 x+1","D"],"edges":["e-0f7951445df45664","e-0cda8cc787155056"],"text":"A081579 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A081579 \u2192 x \u21a6 x+1 \u2192 D","chain":null}