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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":"ffde9ceaf87ea6b51f3358ab2bafe88315ef0660511f2d8aed3d7fd0b10b9674","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-3afaca41e5482e8c","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","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:             47,              24;\n  2:           1633,            2256,             576;\n  3:          50255,          117576,           81216,            13824;\n  4:        1447873,         4824480,         5643648,          2598912,           331776;\n  5:       40017263,       173744760,       289468800,        225745920,         77967360,          7962624;\n  6:     1074869281,      5762485872,     12509622720,      13894502400,      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1104)*x + 576*x^2)*v[i - 1])/(24*n - 1 + 24*x)); 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-9e472291b944d56e","steps":["basis A013609","D+1"],"edges":["e-15af34f262d68ea6","e-bab22f1311280bc8"],"text":"A038326 \u2192 basis A013609 \u2192 D+1"},"headline":"A038326 \u2192 basis A013609 \u2192 D+1","chain":null}