{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","-3"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-16","16","-4","0"],["-12","-4","7","-3"],["-6","-3","-6","0"],["-2","-3","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-16","8","8","-8","2"],["-12","-6","0","1","0"],["-6","-7","-4","0","0"],["-2","-3","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-8","12","-8","2"],["0","-2","-7","4","0"],["0","-4","2","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["8","-8","2"],["2","0","1"],["1","2","0"],["1","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["8","-8","2"],["2","0","1"],["1","2","0"],["1","0","0"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["6","-6","1"],["3","-4","1"],["-2","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["8","-20","17","-3"],["6","7","-19","3"],["0","-6","6","0"],["2","3","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","10","-21","12","-2"],["0","-6","11","-4","0"],["0","4","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["2","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","-3"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","-16","24","-12","2"],["0","-4","2","-2","1"],["0","-2","-3","2","0"],["0","-2","1","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-16","24","-12","2"],["0","-4","2","-2","1"],["0","-2","-3","2","0"],["0","-2","1","0","0"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["-1"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-4","2"],["0","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","-2","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","16","-24","8"],["0","-4","-6","4"],["0","-6","3","0"]]},"deriv_order":0}],"row_multiplier":["0","-2","1"],"start_base":1,"step_multiplier":{"coeffs":[["0","0","4","-4","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","32","-32","8","0"],["0","0","8","4","-10","3"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","16","-24","12","-2"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-4","0"],["-2","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-4","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2"],["1"]]},"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 = phi(n, D) = 1 + (1) D; 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":{"from_index":2,"justification":"the parent theorem (LiuWang) proves P_(m-1) weakly interlacing P_m for every m >= 2; T = phi(n, D) = 1 + (1) D is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 2","kind":"weak","orientation":"unordered","status":"claimed"},"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"derivative_polynomial_real_rooted","statement":"the polynomial in D is real-rooted for every n >= n0","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"}},{"evidence":"delta(n) != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":1,"input_interlacing":{"interlacing":{"from_index":2,"justification":{"premises":["the parent theorem (LiuWang) proves P_(m-1) weakly interlacing P_m for every m >= 2; T = phi(n, D) = 1 + (1) D is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 2"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"proved"},"lean_replay":"not_attempted","operation":{"description":"phi(n, D) = 1 + (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","-3"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D, for every n >= 1"},"parent_proof":{"route":"liu_wang","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":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 2"},"hypotheses":[{"evidence":"f(n) = n + 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n + 1 at n = 2 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["3","1"]},"numerator":["1","1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 2","status":{"kind":"verified"}},{"evidence":"P_1 has degree 0; checked exactly","name":"seed_roots_negative","statement":"P_1 is nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"lambda_n = (n + 1)/(n + 1) > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of lambda_n = (n + 1)/(n + 1) at n = 2 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["3","1"]},"numerator":["1","1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] a_n + (deg P_(n-1)) [x^2] b_n > 0 for every n >= 2 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1(n, x) is the zero polynomial","name":"derivative_coefficient_vanishes_at_zero","statement":"b_n(0) = 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"b_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2 + 4 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = (1)/(n + 1) >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = (1)/(n + 1) at n = 2 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"b_n(x) <= 0 for every x < 0 and every n >= 2","status":{"kind":"verified"}},{"evidence":"a_n(0) = (2 n + 2)/(n + 1) > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of a_n(0) = (2 n + 2)/(n + 1) at n = 2 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1","1"],"method":{"kind":"taylor_shift","shifted_denominator":["3","1"],"shifted_numerator":["6","2"]},"numerator":["2","2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"a_n(0) > 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"exact","name":"requested_root_region","statement":"the requested region (-inf, 0) contains (-inf, 0), the region this theorem certifies","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((2 n + 2)/(n + 1)) + ((3 n - 3)/(n + 1)) x and b_n(x) = ((0)/(n + 1)) + ((4)/(n + 1)) x + ((-2)/(n + 1)) x^2","recurrence":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","-3"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-2"]]},"deriv_order":1,"offset":1,"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 g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang"},"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":"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":"P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((2 n + 2)/(n + 1)) + ((3 n - 3)/(n + 1)) x and b_n(x) = ((0)/(n + 1)) + ((4)/(n + 1)) x + ((-2)/(n + 1)) x^2: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 2","route":"liu_wang","success":true,"variant":"liu_wang"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((2 n + 2)/(n + 1)) + ((3 n - 3)/(n + 1)) x and b_n(x) = ((0)/(n + 1)) + ((4)/(n + 1)) x + ((-2)/(n + 1)) x^2","every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 2"],"theorem":"liu_wang: 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 g = f' (as in their Corollary 2.4), 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","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 = phi(n, D) = 1 + (1) D 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"}