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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"8f7f629181b16ecad63f8c27e91dc2fd9ed14e8b605af7fed68bd179e77124d2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-02156bdfcfec6826","role":"sequence","name":"Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","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:    12,      18;\n  1:    33,     198,        81;\n  2:    78,    1188,      2106,         324;\n  3:   171,    5436,     24462,       18468,         1215;\n  4:   360,   21438,    195696,      385884,       145800,         4374;\n  5:   741,   77274,   1265139,     5060556,      5215995,      1080378,         15309;\n  6:  1506,  262944,   7146954,    50605560,    107204310,     63895392,       7685118,        52488;\n  7:  3039,  861120, 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((n + 1)*x - (3*n + 3)*x^2)*D[P[n - 1, x], x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 12 + 18*x; v[2] = 33 + 198*x + 81*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (6*n^2 + 12*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*v[i - 2] - ((n + 2)*x + (3*n^2 + 6*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*deriv(v[i - 2], x) + (3*n + 4 + (6*n^2 + 12*n + 6)*x)*v[i - 1] + ((n + 1)*x - (3*n + 3)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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-02156bdfcfec6826","steps":["D"],"edges":["e-c97984773cd13005"],"text":"A156366 \u2192 D"},"headline":"A156366 \u2192 D","chain":null}