{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["1","12","1"],["1","45","45","1"],["1","112","420","112","1"]],"recurrence":{"denominator":{"coeffs":[["1"],["3"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","1"],["-11","37"],["30","-24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","48","-48"],["0","-54","54"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-12","24","36"],["34","-124","-14"],["-36","144","-36"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-48","0","48"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","36","36","12"],["-20","-60","-60","-20"],["8","24","24","8"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m)."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-24","-72","-72","-24"],["-28","-84","-84","-28"],["-8","-24","-24","-8"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["234","-948","330"],["182","-740","230"],["36","-144","36"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","48","0","-48"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["-238","104"],["-169","107"],["-30","24"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","114","-114"],["0","54","-54"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["28"],["15"],["2"]]},"deriv_order":0,"shift":3}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2,3],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["14","14","1"],["92","138","48","1"],["646","1292","762","116","1"]],"recurrence":{"denominator":{"coeffs":[["1"],["3"],["2"]]},"terms":[{"coeff":{"coeffs":[["96","144","72","12"],["-160","-240","-120","-20"],["64","96","48","8"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["48","96","36"],["-104","-152","-14"],["72","72","-36"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","96","144","48"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"],["26","37"],["6","-24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-48","-48"],["0","54","54"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["28","15","2"],"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["1","12","1"],["1","45","45","1"],["1","112","420","112","1"]],"recurrence":{"denominator":{"coeffs":[["1"],["3"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","1"],["-11","37"],["30","-24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","48","-48"],["0","-54","54"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-12","24","36"],["34","-124","-14"],["-36","144","-36"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-48","0","48"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","36","36","12"],["-20","-60","-60","-20"],["8","24","24","8"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":3,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["28"],["15"],["2"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-24","-72","-72","-24"],["-28","-84","-84","-28"],["-8","-24","-24","-8"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["234","-948","330"],["182","-740","230"],["36","-144","36"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","48","0","-48"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["-238","104"],["-169","107"],["-30","24"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","114","-114"],["0","54","-54"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["28"],["15"],["2"]]},"deriv_order":0,"shift":3}]},"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 = P((1) x + (1)); so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 3 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"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":1,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (1))","kind":"preserver_image","operator":{"intercept":"1","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1","12","1"],["1","45","45","1"],["1","112","420","112","1"]],"recurrence":{"denominator":{"coeffs":[["1"],["3"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","1"],["-11","37"],["30","-24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","48","-48"],["0","-54","54"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-12","24","36"],["34","-124","-14"],["-36","144","-36"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-48","0","48"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","36","36","12"],["-20","-60","-60","-20"],["8","24","24","8"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)), for every n >= 1"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 1, 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 >= 1, is zero or real-rooted with roots in (-inf, -1]","input_root_region":{"upper":{"closed":true,"value":"-1"}},"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":"Unresolved: row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 1, with p = (-1) (k - 1/2) (k) (k), q = (-1) (k - n - 2) (k - n - 2) (k - n - 3/2); valuation v(n) = 0, degree d(n) = n + 1; 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":"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-1) in the coefficient of x^(s+0) is not shown to be >= 0 for n >= 4","route":"attestation","success":false,"variant":"oeis_catalog:A155495:lean_proved"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 1, 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 = P((1) x + (1)) for every n >= 1 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}