{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["3","5","1"]],"recurrence":{"denominator":{"coeffs":[["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","3"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1","0"],["-3","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","4","-2"],["1","-2","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle for first differences of Catalan numbers."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-1","4","-6","4","-1"],["-1","4","-6","4","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["8","-17","13","-7","3"],["3","-6","4","-2","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-13","17","5","-9"],["-3","4","1","-2"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["6","-12","6"],["1","-2","1"]]},"deriv_order":0,"shift":3}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["2","3"],["3","11","9"]],"recurrence":{"denominator":{"coeffs":[["3","6","3"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["-2","-6","-6","-2"],["1","3","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1","5","9","7","2"],["-3","-12","-19","-14","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","15","18","7"],["3","11","13","5"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["6","1"],"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["3","5","1"]],"recurrence":{"denominator":{"coeffs":[["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","3"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1","0"],["-3","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","4","-2"],["1","-2","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":3,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["6","-12","6"],["1","-2","1"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","2","-1"],["-1","2","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["8","-1","3"],["3","0","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-13","-9"],["-3","-2"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["6"],["1"]]},"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 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0); so the candidate rows satisfy the input rule at every base n >= 0 (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 >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.splits_fPolynomial"],"lean_route":"fTransform","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":"((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0)","kind":"preserver_image","operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["3","5","1"]],"recurrence":{"denominator":{"coeffs":[["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","3"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1","0"],["-3","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","4","-2"],["1","-2","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0), for every n >= 0"},"parent_proof":{"route":"hypergeometric_row","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in [-1, 0]","input_root_region":{"lower":{"closed":true,"value":"-1"},"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":"proved"},"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":"row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 0, with p = (1/2) (k - 2 n - 3) (k) (k + 2), q = (-1/2) (k - 2 n - 2) (k - n - 2) (k - n - 1); valuation v(n) = 0, degree d(n) = n: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0): for n >= 0 the coefficients U_j (U_j = [x^(d(n) - j)] P_n, d(n) = n) satisfy U_j = U_0(n) prod_(t <= j) R(n, t), R(n, j) = (1) (j + n + 2) (j - n - 1) (j - n - 3) / ((j + n + 1) (j + 1) (j)), so sum_j U_j y^j is U_0(n) times the image of (1 + y)^(n) under a nonnegative multiplier sequence, and U_0(n) != 0 for n >= 0 by its scalar recurrence","route":"hypergeometric_row","success":true,"variant":"hypergeometric_row"}],"conditional_on":[],"route":"hypergeometric_row","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0) for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in [-1, 0]"},"kind":"preserver_image"}