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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":"fa4d8a9be80b8c43a66614432c5e9aea46aa22569e7c242e783a05a37e5d9bd5","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-623036dc25b33b8c","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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:            2;\n  2:           20,            24;\n  3:          236,           648,            432;\n  4:         3716,         16568,          24192,          11520;\n  5:        77936,        488048,        1132632,        1152000,          432000;\n  6:      2128544,      17283680,       55664928,       88771392,        69984000,        21772800;\n  7:     73870368,     739623648,     3066112032,     6730684992,      8243533440,    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3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (8*n^3 - 8*n)*x)*v[i - 1])/(4*n - 4)); 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-ecd77b030b8fab94","steps":["basis A013609","D"],"edges":["e-42bb566932c0b2ac","e-8aa07bb5a225dec3"],"text":"A128813 \u2192 basis A013609 \u2192 D"},"headline":"A128813 \u2192 basis A013609 \u2192 D","chain":null}