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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":"c2ed7bdc4eba6ea25fbe0ffcc7364d4b60602fbeb08193ece105dff652c8e4ce","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-a6229b8a5ae7f968","role":"sequence","name":"basis_map_A013609 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:          4,           2;\n  2:         28,          44,           12;\n  3:        248,         744,          632,          120;\n  4:       2656,       12224,        19280,        11488,          1680;\n  5:      33472,      210944,       502400,       547744,        253824,         30240;\n  6:     486144,     3919616,     12664000,     20645184,      17353984,       6612288,        665280;\n  7:    8004096,    78956032,    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1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (32*n^3 - 96*n^2 + 88*n - 24)*x + (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (16*n^2 - 32*n + 12)*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 A013609","D+1"],"edges":["e-17d7b4a637c00413","e-c165df770d145452"],"text":"A109692 \u2192 basis A013609 \u2192 D+1"},"headline":"A109692 \u2192 basis A013609 \u2192 D+1","chain":null}