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n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: coefficient of P_(n-1) must be 1 after normalization; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"HypothesisFailed: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 2, with a_n = 0, b_n = 2, c_n = 1; Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 2: f(n) a_n = 0 = 0 there (exact); first_coefficient_positive: a_2 or c_2 is undefined or a_2 = 0; recurrence_coefficient_positive: fails at n = 3: the denominator of beta_n = 1 vanishes there; root_upper_bound: not attempted: the real-rootedness hypotheses do not all hold","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":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (2) x, V_n(x) = 0, W_n(x) = (-1) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 2: mu_n = -1 = -1 there (exact); constant_term_positive: fails at n = 2: A_n(0) = 0 = 0 there (exact)","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":"the parent equals the flagged entry, but the flag gives real-rootedness only and one-signed coefficients are not shown: the weight of the coefficient of x^s of P_(n-2) in the coefficient of x^(s+2) is not shown to be >= 0 for n >= 2","route":"attestation","success":false,"variant":"oeis_catalog:A132774:lean_proved"},{"detail":"no lean_proved catalog entry equals the parent; 3 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A029600:lean_proved, oeis_catalog:A124932:lean_proved, oeis_catalog:A347823: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 = phi(n, D) = (1) D 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"}