{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["8"],["-16"],["8"]]},"terms":[{"coeff":{"coeffs":[["8","30","18"],["-16","-54","-30"],["8","24","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","22","42"],["0","-4","-22","-24"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","3","12"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"reversal of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","6","6"],["-8","-24","-12"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0","-18"],["0","4","22","24"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","-3","-12"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["0"],["8"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1","2"]],"recurrence":{"denominator":{"coeffs":[["8"],["-16"],["8"]]},"terms":[{"coeff":{"coeffs":[["14","4"],["-19","-8"],["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","-10","-4"],["0","16","4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","12","3"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","0","8"],"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["8"],["-16"],["8"]]},"terms":[{"coeff":{"coeffs":[["8","30","18"],["-16","-54","-30"],["8","24","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","22","42"],["0","-4","-22","-24"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","3","12"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0"],["0"],["8"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","6","6"],["-8","-24","-12"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0","-18"],["0","4","22","24"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","-3","-12"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["0"],["8"]]},"deriv_order":0,"shift":1}]},"statement":"w(n) E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 0); so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 2 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"lean_route":"reversed","lift_summary":{"hypotheses":[{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"x^(d_n) P(1/x), d_n = 0 + 1 (n - 0)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["8"],["-16"],["8"]]},"terms":[{"coeff":{"coeffs":[["8","30","18"],["-16","-54","-30"],["8","24","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","22","42"],["0","-4","-22","-24"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","3","12"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 0), for every n >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every parent row used is zero or real-rooted with all roots in (-inf, 0] (coefficients of one sign)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically"},"verdict":"open"},"notes":[],"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero constant","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a derivative or a lag term","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + gamma(n) (x - L)(x - U) P'_(n-1)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_d(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a nonzero lag term and a nonzero P_(n-1) or P'_(n-1) term","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"HypothesisFailed: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = 2 n - 3 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 36 - 42 n + 12 n^2 + 14 n x - 8 n^2 x - 4 x^2 + 4 n x^2, B = 54 x - 24 n x + 2 x^2 + 10 n x^2 - 4 x^3 + 4 n x^3, C = 12 x^2 + 3 x^3) / (8 n^2 - 16 n + 8); open hypotheses: weyl_step_discriminant_nonnegative: at n = 2: B^2 - 4 A C = -93057081933695541349658178057759/316912650057057350374175801344 < 0 at x = -202193/131072: T is not a real-rootedness preserver on the whole line (Borcea--Branden 2010, Theorem 1.3, necessary condition); weyl_step_interlacing: at n = 2: A does not strictly interlace B (exact Bezout test)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"Unresolved: row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 1, with p = (-3/16) (k - 4/3 n - 1) (k - 2 n) (k), q = (-3/4) (k - n - 1) (k - n - 1/2) (k - 4/3 n); valuation v(n) = 0, degree d(n) = n; open hypotheses: multiplier_matching: no matching of the ratio factors (identity: R(n, j) = -q(n, j + v(n)) / p(n, j + v(n))) with a base binom(M, j), complements, pairs and reciprocal rising factorials holds for every n >= 1 + delay, delay <= 4, also after the reversal at the degree","route":"hypergeometric_row","success":false,"variant":"hypergeometric_row"},{"detail":"no lean_proved catalog entry equals the parent; 25 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A029653:lean_proved, oeis_catalog:A060920:lean_proved, oeis_catalog:A084950:lean_proved, oeis_catalog:A093375:lean_proved, oeis_catalog:A104029:lean_proved, oeis_catalog:A106246:lean_proved, oeis_catalog:A113214:lean_proved, oeis_catalog:A113413:lean_proved, oeis_catalog:A117317:lean_proved, oeis_catalog:A119308:lean_proved, oeis_catalog:A119808:lean_proved, oeis_catalog:A121574:lean_proved, oeis_catalog:A126216:lean_proved, oeis_catalog:A133087:lean_proved, oeis_catalog:A134082:lean_proved, oeis_catalog:A154929:lean_proved, oeis_catalog:A156319:lean_proved, oeis_catalog:A167194:lean_proved, oeis_catalog:A193727:lean_proved, oeis_catalog:A193728:lean_proved, oeis_catalog:A193730:lean_proved, oeis_catalog:A193735:lean_proved, oeis_catalog:A202241:lean_proved, oeis_catalog:A318397:lean_proved, oeis_catalog:A394813:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 0) for every n >= 0 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}