{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"reversal"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["8"],["-10"],["4"]]},"terms":[{"coeff":{"coeffs":[["-2","-52"],["8","152"],["-10","-132"],["4","36"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-62"],["0","-18","134"],["0","8","-52"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","3","-12"],["0","0","-5","20"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"theta_plus_one of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","-4"],["0","4"],["-2","24"],["-4","-36"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","2","-20"],["0","2","-30"],["0","-8","52"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","2","-8"],["0","0","5","-20"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["0"],["2"],["4"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["8"],["-10"],["4"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["2","7"],["-6","-16"],["4","7"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-10","10"],["0","10","-18"],["0","12","2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-12","3"],["0","0","20","-5"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","0","2","4"],"operator":{"schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"reversal"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["8"],["-10"],["4"]]},"terms":[{"coeff":{"coeffs":[["-2","-52"],["8","152"],["-10","-132"],["4","36"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-62"],["0","-18","134"],["0","8","-52"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","3","-12"],["0","0","-5","20"]]},"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"],["2"],["4"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-4"],["0","4"],["-2","24"],["-4","-36"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","2","-20"],["0","2","-30"],["0","-8","52"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","2","-8"],["0","0","5","-20"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["0"],["2"],["4"]]},"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 - 1); so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 1 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":"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":1,"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 - 1)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["8"],["-10"],["4"]]},"terms":[{"coeff":{"coeffs":[["-2","-52"],["8","152"],["-10","-132"],["4","36"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-62"],["0","-18","134"],["0","8","-52"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","3","-12"],["0","0","-5","20"]]},"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 - 1), for every n >= 1"},"parent_proof":{"route":"hypergeometric_row","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, 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":"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 theorem needs at least two seed rows","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 theorem needs two seed rows","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 theorem needs two seed rows","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"HypothesisFailed: P_n = a_n(x) S_n(P_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = -2 + 8 n - 10 n^2 + 4 n^3 - 52 x + 152 n x - 132 n^2 x + 36 n^3 x, B = 8 x - 18 n x + 8 n^2 x - 62 x^2 + 134 n x^2 - 52 n^2 x^2, C = 3 x^2 - 5 n x^2 - 12 x^3 + 20 n x^3) / (4 n^3 - 10 n^2 + 8 n - 2); open hypotheses: weyl_step_discriminant_nonnegative: at n = 2: B^2 - 4 A C = -95818514256575/17179869184 < 0 at x = -781/512: T is not a real-rootedness preserver on the whole line (Borcea--Branden 2010, Theorem 1.3, necessary condition); weyl_step_leading_coefficients_same_sign: fails at n = 2: A_1 B_lead = -1872 n^5 + 11688 n^4 - 27824 n^3 + 31256 n^2 - 16392 n + 3224 = -24 there (exact); weyl_step_interlacing: fails at n = 2: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = 103168 n^10 - 1097856 n^9 + 5132416 n^8 - 13884192 n^7 + 24074192 n^6 - 27961440 n^5 + 22034704 n^4 - 11635968 n^3 + 3942160 n^2 - 774144 n + 66960 = 720 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 2: A_a B_(a+1) = -1872 n^5 + 11688 n^4 - 27824 n^3 + 31256 n^2 - 16392 n + 3224 = -24 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 1, with p = (1/4) (k - 2 n) (k) (k), q = (-1) (k - n - 1/2) (k - n) (k - n); valuation v(n) = 0, degree d(n) = n - 1: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0): for n >= 1 the coefficients U_j (U_j = [x^(d(n) - j)] P_n, d(n) = n - 1) satisfy U_j = U_0(n) prod_(t <= j) R(n, t), R(n, j) = (1/4) (j + n) (j - n) (j - n) / ((j + 1/2) (j) (j)), so sum_j U_j y^j is U_0(n) times the image of (1 + y)^(n - 1) under a nonnegative multiplier sequence, and U_0(n) != 0 for n >= 1 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 >= 1, 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 = x^(d_n) P(1/x), d_n = 0 + 1 (n - 1) for every n >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}