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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":"fe53c4e97f62ab805c024cf8fea7e25ca53d7e00ae6338d93d19757b309de56d","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-a5662819bbc56ccb","role":"sequence","name":"gamma_polynomial of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:    2;\n  2:    6,    2;\n  3:   16,    8;\n  4:   40,   32,    4;\n  5:   96,  112,   24;\n  6:  224,  352,  120,    8;\n  7:  512, 1024,  512,   64;\n  8: 1152, 2816, 1920,  384,  16;\n  9: 2560, 7424, 6528, 1920, 160;\n","recurrence_sentence":"The row polynomials satisfy (n^3-n^2+(4*n^2-8*n)*x+4*n*x^2)*P(n,x) = (2*n^3-2*n+(2*n^3+4*n^2-14*n)*x+(8*n^2-8*n+8)*x^2+(8*n-8)*x^3)*P(n-1,x) - ((2*n^2+2*n)*x+(4*n^2+12*n)*x^2+(16*n+8)*x^3+16*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[((2*n^3 - 2*n + (2*n^3 + 4*n^2 - 14*n)*x + (8*n^2 - 8*n + 8)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - ((2*n^2 + 2*n)*x + (4*n^2 + 12*n)*x^2 + (16*n + 8)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - n^2 + (4*n^2 - 8*n)*x + 4*n*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] = ((2*n^3 - 2*n + (2*n^3 + 4*n^2 - 14*n)*x + (8*n^2 - 8*n + 8)*x^2 + (8*n - 8)*x^3)*v[i - 1] - ((2*n^2 + 2*n)*x + (4*n^2 + 12*n)*x^2 + (16*n + 8)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - n^2 + (4*n^2 - 8*n)*x + 4*n*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-1bffaa6198344f93","steps":["\u03b3","D+1"],"edges":["e-babfdb0b6fa40ea8","e-730a03d25af5b037"],"text":"A059473 \u2192 \u03b3 \u2192 D+1"},"headline":"A059473 \u2192 \u03b3 \u2192 D+1","chain":null}