{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"parent":{"name":"Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","3"],["6","12","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","3","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"symmetry_schedule":{"first_degree":0,"first_index":0,"step":1}},"scale":"1"},"gamma_rule":"P(n) = 3 P(n-1) - 3 P(n-2) + P(n-3)","identifying_rows":3,"identity":{"certificate":{"candidate_leader_shift":3,"candidate_relation":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-3"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":3}]},"candidate_valid_from_base":0,"exceptional_bases":[],"gamma_relation":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-3"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":3}]},"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["3"],["6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"parent":{"first_index":0,"initial_polynomials":[["1"],["3","3"],["6","12","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","3","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"quotients":[{"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":0}],"scale":"1","statement":"every parent row P_m, m >= 0, is symmetric at d_m; the gamma rows satisfy the transported rule P(n) = 3 P(n-1) - 3 P(n-2) + P(n-3); w(n) M = sum_nu H_nu S^nu L with remainder zero for the relation L of H_n = lambda(n) Gamma_(n+0)(c z), c = 1, valid from base 0; with the exceptional bases checked exactly H satisfies the input rule at every base n >= 0, and agreement on the 3 identifying rows gives Q_n = H_n for every n >= 0","symmetry_schedule":{"first_degree":0,"first_index":0,"step":1}},"status":"certified"},"input_first_index":0,"lift_summary":{"hypotheses":[{"evidence":"the reversal transport x^(d_m) P_m(1/x) of the parent rule defines the same row map, both left factors are nonzero polynomials at every target m >= 3, and the seeds P_0..P_2 are symmetric; so the reversed rows coincide with the rows by induction and every P_m, m >= 0, is symmetric at d_m","name":"parent_rows_symmetric","statement":"every parent row P_m, m >= p0, is symmetric at d_m","status":{"kind":"verified"}},{"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":"no published theorem transfers interlacing of consecutive parent rows to their gamma polynomials under hypotheses polytool checks","status":"not_proved"},"lean_replay":"not_attempted","operation":{"kind":"gamma_polynomial","scale":"1","symmetry_schedule":{"first_degree":0,"first_index":0,"step":1}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","3"],["6","12","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","3","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) Gamma_(n+0)(c z) with c = 1 and P_m = sum_j Gamma_(m,j) x^j (1 + x)^(d_m - 2j), 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), and symmetric at its index d_m","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c"},"verdict":"open"},"notes":[],"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of 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":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 3 row(s)"},{"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":"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+0) is not shown to be >= 0 for n >= 3","route":"attestation","success":false,"variant":"oeis_catalog:A094305:lean_proved"}],"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"},"parent_symmetry":{"detail":"the reversal transport x^(d_m) P_m(1/x) of the parent rule defines the same row map, both left factors are nonzero polynomials at every target m >= 3, and the seeds P_0..P_2 are symmetric; so the reversed rows coincide with the rows by induction and every P_m, m >= 0, is symmetric at d_m","status":"certified"},"rows_compared":16,"schema":"polytool.recurrence-gamma-lift.v1","statement":"Q_n = lambda(n) Gamma_(n+0)(c z), c = 1 for every n >= 0 is certified; the lift is not proved (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":"gamma_polynomial"}