{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1","10","15"]],"recurrence":{"denominator":{"coeffs":[["12"],["14"],["4"]]},"terms":[{"coeff":{"coeffs":[["12","45"],["14","57"],["4","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","13","15"],["0","8","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows: T(n, k) = binomial(n, k)*binomial(2*n+k, k), 0 <= k <= n."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-42","-144"],["-26","-102"],["-4","-18"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-22","-24"],["0","-8","-9"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["20"],["18"],["4"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["10","30"]],"recurrence":{"denominator":{"coeffs":[["6"],["10"],["4"]]},"terms":[{"coeff":{"coeffs":[["20","60"],["18","66"],["4","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","14","15"],["0","8","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["30","22","4"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1","10","15"]],"recurrence":{"denominator":{"coeffs":[["12"],["14"],["4"]]},"terms":[{"coeff":{"coeffs":[["12","45"],["14","57"],["4","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","13","15"],["0","8","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","20","20"],["0","18","18"],["0","4","4"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["-10"],["-9"],["-2"]]},"deriv_order":0},{"coeff":{"coeffs":[["1","4"],["0","1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","-240","-240"],["0","-186","-186"],["0","-36","-36"]]},"deriv_order":0}],"row_multiplier":["0","1","1"],"start_base":0,"step_multiplier":{"coeffs":[["0","0","1","2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","51","294","435","192"],["0","0","30","180","270","120"],["0","0","4","26","40","18"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","21","66","69","24"],["0","0","0","8","25","26","9"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-30","-120"],["-22","-93"],["-4","-18"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-21","-24"],["0","-8","-9"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["30"],["22"],["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 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = phi(n, D) = (1) D; so the candidate rows satisfy the input rule at every base n >= 0 (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 >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"none"},"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 [0, 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 [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + 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":"0","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":0,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"phi(n, D) = (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1","10","15"]],"recurrence":{"denominator":{"coeffs":[["12"],["14"],["4"]]},"terms":[{"coeff":{"coeffs":[["12","45"],["14","57"],["4","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","13","15"],["0","8","9"]]},"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) D, for every n >= 0"},"parent_proof":{"route":"iterated_preserver","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":"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_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":"Unresolved: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = ((n^2 + 7/2 n + 3)/(n^2 + 7/2 n + 3)) + ((9/2 n^2 + 57/4 n + 45/4)/(n^2 + 7/2 n + 3)) x and b_n(x) = ((0)/(n^2 + 7/2 n + 3)) + ((2 n + 13/4)/(n^2 + 7/2 n + 3)) x + ((9/4 n + 15/4)/(n^2 + 7/2 n + 3)) x^2; open hypotheses: derivative_coefficient_nonpositive: b_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] b_n <= 0 fails for k = 2 (not a disproof): fails at n = 1: [x^2] b_n = 4/5 there (exact)","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): c_1 is not gamma(n) w(x)","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":"P_n = a_n(x) S_n(P_(n-1)) for n >= 1, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 12 + 14 n + 4 n^2 + 45 x + 57 n x + 18 n^2 x, B = 13 x + 8 n x + 15 x^2 + 9 n x^2, C = 0) / (4 n^2 + 14 n + 12): every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 1","route":"iterated_preserver","success":true,"variant":"iterated_preserver"}],"conditional_on":[],"route":"iterated_preserver","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 = phi(n, D) = (1) D 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"}