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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":"255d61a8d08f2a0a5cae2aeb98e847230f9de6df4154d571a79df65ae80b48d4","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-3bc26d3108ee6bfd","role":"sequence","name":"Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.","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:        1;\n  2:       14,          2;\n  3:      111,        222,            3;\n  4:      796,       5772,         2388,            4;\n  5:     5597,     105284,       157926,        22388,            5;\n  6:    39210,    1648542,      6001188,      3297084,       196050,            6;\n  7:   274507,   23862066,    175842897,    234457196,     59655165,      1647042,           7;\n  8:  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{n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(42*n - 35 + (72*n^2 - 264*n + 170)*x + (216*n^3 - 1188*n^2 + 1962*n - 935)*x^2)*v[i - 2] - ((36*n - 30)*x + (216*n^2 - 612*n + 360)*x^2 - (216*n^2 - 576*n + 330)*x^3)*deriv(v[i - 2], x) + (48*n - 82 + (72*n^2 - 264*n + 242)*x)*v[i - 1] + ((36*n - 66)*x - (36*n - 66)*x^2)*deriv(v[i - 1], x))/(6*n - 11)); 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-3bc26d3108ee6bfd","steps":["D"],"edges":["e-09f3b3f8d3a2391e"],"text":"A142461 \u2192 D"},"headline":"A142461 \u2192 D","chain":null}