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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":"9f1745ceaa4e287bf17f47c3891d77b80d396d21a4fe6ff0c972bdee1cf9a474","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-d0986a7acf60271a","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:    1;\n  1:    1;\n  2:    3,     1;\n  3:   10,     7,     1;\n  4:   32,    35,    11,     1;\n  5:   99,   149,    75,    15,    1;\n  6:  299,   576,   405,   130,   19,    1;\n  7:  887,  2088,  1899,   854,  200,   23,   1;\n  8: 2595,  7229,  8097,  4725, 1550,  285,  27,  1;\n  9: 7508, 24179, 32243, 23243, 9900, 2547, 385, 31, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-9*n^2+11*n-3-(8*n-12)*x-(2*n-3)*x^2)*P(n,x) = (3*n^3-7*n^2-4*n+9+(n^3-2*n^2-18*n+23)*x-(8*n-9)*x^2-(n-1)*x^3)*P(n-1,x) + (5*n^2-10*n-30+(6*n^2-12*n-66)*x+(n^2-2*n-47)*x^2-12*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 7*n^2 - 4*n + 9 + (n^3 - 2*n^2 - 18*n + 23)*x - (8*n - 9)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (5*n^2 - 10*n - 30 + (6*n^2 - 12*n - 66)*x + (n^2 - 2*n - 47)*x^2 - 12*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 9*n^2 + 11*n - 3 - (8*n - 12)*x - (2*n - 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^3 - 7*n^2 - 4*n + 9 + (n^3 - 2*n^2 - 18*n + 23)*x - (8*n - 9)*x^2 - (n - 1)*x^3)*v[i - 1] + (5*n^2 - 10*n - 30 + (6*n^2 - 12*n - 66)*x + (n^2 - 2*n - 47)*x^2 - 12*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 9*n^2 + 11*n - 3 - (8*n - 12)*x - (2*n - 3)*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-01fe968f3e05a927","steps":["x \u21a6 x+1","D+1"],"edges":["e-17055ffc05dec6ce","e-1ac18839ef8b8a43"],"text":"A165253 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A165253 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}