{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-16","-8"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","5"],["0","0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","3"],["0","0","2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-6","-4"],["-16","-8"],["-8","-4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["2"],["5"],["2"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-2"],["5"],["-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","6"],["-8","-16"],["4","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["2","5","2"],"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-16","-8"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","5"],["0","0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["2"],["5"],["2"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","3"],["0","0","2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-6","-4"],["-16","-8"],["-8","-4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["2"],["5"],["2"]]},"deriv_order":0,"shift":2}]},"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 = x^(d_n) P(1/x), 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 2 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.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":0,"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 - 0)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-16","-8"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","5"],["0","0","-2"]]},"deriv_order":0,"offset":2,"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 - 0), for every n >= 0"},"parent_proof":{"route":"liu_wang_derivative_lag","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 (-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_interlacing_report":{"conclusion":{"first_index":0,"interlacing":"weak","interlacing_from":1,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); P_n = x^1 Q_n for n >= 1 with Q_n(0) != 0, and for n >= 2 the Q_n have simple roots, Q_(n-1) strictly interlaces Q_n, and P_(n-1) interlaces P_n (Liu--Wang, common root 0); the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 1, ..., 1"},"hypotheses":[{"evidence":"f(n) = 2 n^2 - 3 n != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = 2 n^2 - 3 n at n = 3 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["9","9","2"]},"numerator":["0","-3","2"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 3","status":{"kind":"verified"}},{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_1 / x^1 and P_2 / x^1 are nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"degrees 0 and 1, leading coefficients of one sign","name":"seed_degree_step","statement":"deg P_2 / x^1 = deg P_1 / x^1 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_1 / x^1 strictly interlaces P_2 / x^1 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_0","name":"earlier_seeds","statement":"the seed rows P_0, ..., P_0 are each zero or real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"shared degree-law certificate, route recurrence (RealRooted RealRooted.LinRec.natDegree_eq_and_leadingCoeff_pos_of_ratio, RealRooted.ScalarRec.pos_of_ratio): the degree law deg P_n = 0 + 1 (n - 0) with positive leading coefficient for n >= 0 (route recurrence, induction from n = 3); every row is nonzero; degree steps {1} from n = 2; the certificate is recomputed when the report is replayed","name":"degree_law","statement":"deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign for every n >= 2","status":{"kind":"verified"}},{"evidence":"[x^0] c_1 and [x^0] c_2 are zero polynomials","name":"multipliers_vanish_at_zero","statement":"V_n(0) = 0 and W_n(0) = 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"A_n(0) = (4 n^2 - 8 n + 3)/(n^2 - 3/2 n) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of A_n(0) = (4 n^2 - 8 n + 3)/(n^2 - 3/2 n) at n = 3 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["0","-3/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["9/2","9/2","1"],"shifted_numerator":["15","16","4"]},"numerator":["3","-8","4"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"V_n = 0","name":"derivative_coefficient_nonpositive","statement":"V_n(x) <= 0 for every x < 0 and every n >= 3","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (1/2 n^2 - 5/4 n + 1/2)/(n^2 - 3/2 n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (1/2 n^2 - 5/4 n + 1/2)/(n^2 - 3/2 n) at n = 3 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["0","-3/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["9/2","9/2","1"],"shifted_numerator":["5/4","7/4","1/2"]},"numerator":["1/2","-5/4","1/2"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"W_n(x) <= 0 for every x < 0 and every n >= 3","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (1/2 n^2 - 5/4 n + 1/2)/(n^2 - 3/2 n) > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (1/2 n^2 - 5/4 n + 1/2)/(n^2 - 3/2 n) at n = 3 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["0","-3/2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["9/2","9/2","1"],"shifted_numerator":["5/4","7/4","1/2"]},"numerator":["1/2","-5/4","1/2"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"for every n >= 3 and x < 0, V_n(x) < 0 or W_n(x) < 0 (V_n < 0 on (-inf, 0) for every n, or W_n < 0 there for every n)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"lean_replay":"not_attempted","lean_route":"sign","normal_form":"P_n = x^1 Q_n for n >= 1 (x^1 divides P_1 and P_2, and V_n(0) = 0), with Q_n = A_n(x) Q_(n-1) + V_n(x) Q'_(n-1) + W_n(x) Q_(n-2) for n >= 3, A_n(x) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) x, V_n(x) = 0, W_n(x) = ((-n^2 + 5/2 n - 1)/(n^2 - 3/2 n)) x^2 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) x)","recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-16","-8"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","5"],["0","0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","schema":"polytool.recurrence-real-rootedness-theorem.v1","start_delay":1,"status":"proved","theorem":"liu_wang_derivative_lag"},"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":"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 = (4 n^2 - 8 n + 3)/(n^2 - 3/2 n), b_n = (2 n^2 - 4 n + 2)/(n^2 - 3/2 n), c_n = (n^2 - 5/2 n + 1)/(n^2 - 3/2 n); 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: seeds_interlace: deg Q_1 != deg Q_0 + 1 = 1; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; 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":"induction start moved 1 row(s) past the seeds: P_n = x^1 Q_n for n >= 1 (x^1 divides P_1 and P_2, and V_n(0) = 0), with Q_n = A_n(x) Q_(n-1) + V_n(x) Q'_(n-1) + W_n(x) Q_(n-2) for n >= 3, A_n(x) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) x, V_n(x) = 0, W_n(x) = ((-n^2 + 5/2 n - 1)/(n^2 - 3/2 n)) x^2 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) x): every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); P_n = x^1 Q_n for n >= 1 with Q_n(0) != 0, and for n >= 2 the Q_n have simple roots, Q_(n-1) strictly interlaces Q_n, and P_(n-1) interlaces P_n (Liu--Wang, common root 0); the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 1, ..., 1","route":"liu_wang_derivative_lag","success":true,"variant":"liu_wang_derivative_lag"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = x^1 Q_n for n >= 1 (x^1 divides P_1 and P_2, and V_n(0) = 0), with Q_n = A_n(x) Q_(n-1) + V_n(x) Q'_(n-1) + W_n(x) Q_(n-2) for n >= 3, A_n(x) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) x, V_n(x) = 0, W_n(x) = ((-n^2 + 5/2 n - 1)/(n^2 - 3/2 n)) x^2 (A_n = a_n + 1 V_n / x for the rule's a_n(x) = ((4 n^2 - 8 n + 3)/(n^2 - 3/2 n)) + ((2 n^2 - 4 n + 2)/(n^2 - 3/2 n)) x)","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); P_n = x^1 Q_n for n >= 1 with Q_n(0) != 0, and for n >= 2 the Q_n have simple roots, Q_(n-1) strictly interlaces Q_n, and P_(n-1) interlaces P_n (Liu--Wang, common root 0); the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly); P_(m-1) interlaces P_m checked exactly for m = 1, ..., 1"],"theorem":"liu_wang_derivative_lag: L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"liu_wang_derivative_lag","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 = x^(d_n) P(1/x), 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 (-inf, 0]"},"kind":"preserver_image"}