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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":"734c0047643d95bc7fc6447bc614892c532d34d320fe3b3ad0c7ff8c72edb7a6","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-261c6497576cc5f7","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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:          3,           1;\n  2:         18,          16,           3;\n  3:        148,         236,         113,          15;\n  4:       1520,        3624,        3032,        1016,         105;\n  5:      18656,       60144,       74132,       42812,       11139,         945;\n  6:     266112,     1089920,     1800472,     1517912,      675154,      144264,       10395;\n  7:    4324608,    21578752,    45029840,    50574192,  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(16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*v[i - 1])/(2*n - 3)); 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-2b477f2aa77e3efc","steps":["basis A007318","D+1"],"edges":["e-07dfe9939e4be29b","e-42e0fc1af5be220a"],"text":"A109692 \u2192 basis A007318 \u2192 D+1"},"headline":"A109692 \u2192 basis A007318 \u2192 D+1","chain":null}