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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":"0f2b97badb6048d78fbd4ceaa77563186703aa8a2f9c18a678b4304751775b54","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-3c0665ea87e1dc8e","role":"sequence","name":"basis_map_A007318 of A Riordan array of coordination sequences.","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:    23,    18,     3;\n  4:    76,    96,    36,     4;\n  5:   233,   408,   246,    60,     5;\n  6:   682,  1530,  1284,   500,    90,     6;\n  7:  1935,  5298,  5721,  3100,   885,   126,    7;\n  8:  5368, 17360, 22968, 16040,  6360,  1428,  168,   8;\n  9: 14641, 54624, 85644, 73432, 37530, 11676, 2156, 216, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-12*n+8+(n^3-2*n^2-4*n+5)*x-(n^2-1)*x^2)*P(n,x) = -(4*n^3-32*n^2+56*n-16+(3*n^3-35*n^2+75*n-16)*x-(7*n^2-27*n+2)*x^2+3*n*x^3)*P(n-1,x) + (8*n^2-24*n+8+(16*n^2-56*n+16)*x+(9*n^2-43*n+9)*x^2+(n^2-12*n+1)*x^3-n*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[(-(4*n^3 - 32*n^2 + 56*n - 16 + (3*n^3 - 35*n^2 + 75*n - 16)*x - (7*n^2 - 27*n + 2)*x^2 + 3*n*x^3)*P[n - 1, x] + (8*n^2 - 24*n + 8 + (16*n^2 - 56*n + 16)*x + (9*n^2 - 43*n + 9)*x^2 + (n^2 - 12*n + 1)*x^3 - n*x^4)*D[P[n - 1, x], x])/(4*n^2 - 12*n + 8 + (n^3 - 2*n^2 - 4*n + 5)*x - (n^2 - 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] = (-(4*n^3 - 32*n^2 + 56*n - 16 + (3*n^3 - 35*n^2 + 75*n - 16)*x - (7*n^2 - 27*n + 2)*x^2 + 3*n*x^3)*v[i - 1] + (8*n^2 - 24*n + 8 + (16*n^2 - 56*n + 16)*x + (9*n^2 - 43*n + 9)*x^2 + (n^2 - 12*n + 1)*x^3 - n*x^4)*deriv(v[i - 1], x))/(4*n^2 - 12*n + 8 + (n^3 - 2*n^2 - 4*n + 5)*x - (n^2 - 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-ec35d1145e9a1211","steps":["x \u21a6 x+1","D"],"edges":["e-0b1886c9648f8656","e-882ea3ea8e99d88b"],"text":"A113413 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A113413 \u2192 x \u21a6 x+1 \u2192 D","chain":null}