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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":"0d2d55fd525b0116d38de88d1ee6e56759c32c1257d940c4faba320f48438121","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-3bdf31527eaa2fb3","role":"sequence","name":"basis_map_A007318 of Pascal-(1,2,1) array read by antidiagonals.","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:     6,      2;\n  3:    24,     20,      3;\n  4:    88,    116,     42,      4;\n  5:   300,    544,    324,     72,     5;\n  6:   984,   2272,   1896,    696,   110,     6;\n  7:  3136,   8800,   9480,   4928,  1280,   156,    7;\n  8:  9792,  32352,  42768,  28944, 10680,  2124,  210,   8;\n  9: 30096, 114432, 179424, 150048, 72420, 20448, 3276, 272, 9;\n","recurrence_sentence":"The row polynomials satisfy (6*n-6+(3*n^2-3*n)*x-(n-1)*x^2)*P(n,x) = -(18*n^2-78*n+48+(15*n^2-87*n+66)*x-(14*n-22)*x^2+2*x^3)*P(n-1,x) + (36*n-36+(72*n-84)*x+(39*n-63)*x^2+(3*n-16)*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[(-(18*n^2 - 78*n + 48 + (15*n^2 - 87*n + 66)*x - (14*n - 22)*x^2 + 2*x^3)*P[n - 1, x] + (36*n - 36 + (72*n - 84)*x + (39*n - 63)*x^2 + (3*n - 16)*x^3 - x^4)*D[P[n - 1, x], x])/(6*n - 6 + (3*n^2 - 3*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] = (-(18*n^2 - 78*n + 48 + (15*n^2 - 87*n + 66)*x - (14*n - 22)*x^2 + 2*x^3)*v[i - 1] + (36*n - 36 + (72*n - 84)*x + (39*n - 63)*x^2 + (3*n - 16)*x^3 - x^4)*deriv(v[i - 1], x))/(6*n - 6 + (3*n^2 - 3*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-f25cd955a876cde3","steps":["x \u21a6 x+1","D"],"edges":["e-6f8e74d475c3b6cc","e-32cc03140fae5ebc"],"text":"A081577 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A081577 \u2192 x \u21a6 x+1 \u2192 D","chain":null}