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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":"06d3e6e0c3b1a040865dde9a90f39765c7640cb1f2bbabf57fe04a6d17fc134d","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-ee6acb764200c6c7","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 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:       28,          8;\n  3:      244,        488,           24;\n  4:     1844,      11712,         5532,           64;\n  5:    13260,     203240,       304860,        53040,          160;\n  6:    93684,    3022320,     10365240,      6044640,       468420,          384;\n  7:   657836,   41326776,    275945640,    367927520,    103316940,      3947016,         896;\n  8:  4609588,  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// Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n - 42 + (100*n^2 - 320*n + 156)*x + (125*n^3 - 600*n^2 + 835*n - 312)*x^2)*v[i - 2] - ((50*n - 30)*x + (125*n^2 - 325*n + 150)*x^2 - (125*n^2 - 275*n + 120)*x^3)*deriv(v[i - 2], x) + (45*n - 62 + (50*n^2 - 160*n + 128)*x)*v[i - 1] + ((25*n - 40)*x - (25*n - 40)*x^2)*deriv(v[i - 1], x))/(5*n - 8)); 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-ee6acb764200c6c7","steps":["D"],"edges":["e-6f9f2e4e97e7a43c"],"text":"A257614 \u2192 D"},"headline":"A257614 \u2192 D","chain":null}