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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":"17968f90156b46e07ea3e7c5c9be62bafa4accc09400e9bb28f3919eca85ce0f","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-4438653f9e73a43c","role":"sequence","name":"reversal of Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-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:   2;\n  2:   5,    10;\n  3:  10,    46,     45;\n  4:  19,   152,    327,     208;\n  5:  36,   448,   1593,    2176,    1015;\n  6:  69,  1262,   6612,   14564,   14380,    5262;\n  7: 134,  3498,  25413,   80356,  126040,   96678,   28980;\n  8: 263,  9676,  93993,  400684,  888155, 1071384,  668465,  169176;\n  9: 520, 26908, 341421, 1887308, 5547365, 9358728, 9117157, 4777240, 1043775;\n","recurrence_sentence":"The row polynomials satisfy (n-4+(2*n-4)*x)*P(n,x) = -(2*n-6-(2*n^2-22*n+40)*x+(n^3-18*n^2+90*n-111)*x^2+(4*n^3-44*n^2+146*n-130)*x^3+(5*n^3-40*n^2+97*n-66)*x^4+(2*n^3-12*n^2+22*n-12)*x^5)*P(n-2,x) - ((n-3)*x-(n^2-11*n+20)*x^2-(5*n^2-34*n+45)*x^3-(9*n^2-45*n+46)*x^4-(7*n^2-27*n+22)*x^5-(2*n^2-6*n+4)*x^6)*d/dx P(n-2,x) + (3*n-11-(2*n^2-24*n+50)*x-(6*n^2-41*n+58)*x^2-(4*n^2-18*n+20)*x^3)*P(n-1,x) + ((n-4)*x+(4*n-12)*x^2+(5*n-12)*x^3+(2*n-4)*x^4)*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[(-(2*n - 6 - (2*n^2 - 22*n + 40)*x + (n^3 - 18*n^2 + 90*n - 111)*x^2 + (4*n^3 - 44*n^2 + 146*n - 130)*x^3 + (5*n^3 - 40*n^2 + 97*n - 66)*x^4 + (2*n^3 - 12*n^2 + 22*n - 12)*x^5)*P[n - 2, x] - ((n - 3)*x - (n^2 - 11*n + 20)*x^2 - (5*n^2 - 34*n + 45)*x^3 - (9*n^2 - 45*n + 46)*x^4 - (7*n^2 - 27*n + 22)*x^5 - (2*n^2 - 6*n + 4)*x^6)*D[P[n - 2, x], x] + (3*n - 11 - (2*n^2 - 24*n + 50)*x - (6*n^2 - 41*n + 58)*x^2 - (4*n^2 - 18*n + 20)*x^3)*P[n - 1, x] + ((n - 4)*x + (4*n - 12)*x^2 + (5*n - 12)*x^3 + (2*n - 4)*x^4)*D[P[n - 1, x], x])/(n - 4 + (2*n - 4)*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] = (-(2*n - 6 - (2*n^2 - 22*n + 40)*x + (n^3 - 18*n^2 + 90*n - 111)*x^2 + (4*n^3 - 44*n^2 + 146*n - 130)*x^3 + (5*n^3 - 40*n^2 + 97*n - 66)*x^4 + (2*n^3 - 12*n^2 + 22*n - 12)*x^5)*v[i - 2] - ((n - 3)*x - (n^2 - 11*n + 20)*x^2 - (5*n^2 - 34*n + 45)*x^3 - (9*n^2 - 45*n + 46)*x^4 - (7*n^2 - 27*n + 22)*x^5 - (2*n^2 - 6*n + 4)*x^6)*deriv(v[i - 2], x) + (3*n - 11 - (2*n^2 - 24*n + 50)*x - (6*n^2 - 41*n + 58)*x^2 - (4*n^2 - 18*n + 20)*x^3)*v[i - 1] + ((n - 4)*x + (4*n - 12)*x^2 + (5*n - 12)*x^3 + (2*n - 4)*x^4)*deriv(v[i - 1], x))/(n - 4 + (2*n - 4)*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-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","reverse","D"],"edges":["e-236d0fb15dc6eca0","e-0bbf9a609051b56f","e-92db6ced2c0b91de"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 reverse \u2192 D"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 reverse \u2192 D","chain":null}