{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-7","-6"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"derivative of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2))."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["40","62","24","0"],["-30","-4","24","4"],["0","-20","-13","2"],["0","0","-3","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["50","160","156","48","0"],["0","20","96","82","12"],["0","0","-10","5","10"],["0","0","0","-3","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","25","60","47","12","0"],["0","0","25","50","29","4"],["0","0","0","5","9","4"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-10","-3","0"],["10","-2","1"],["0","5","-2"],["0","0","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-10","-3","0"],["0","10","-2","1"],["0","0","5","-2"],["0","0","0","1"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-20","4","-4"],["10","-12","8"],["0","5","-5"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-70","-46","10","0"],["30","-36","-41","6"],["0","20","4","-8"],["0","0","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-25","-35","-2","8","0"],["0","-25","-40","-11","4"],["0","0","-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":2,"multiplier":["-1","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-7","-6"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":2,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["-25/2","-105/4","-67/4","-3","0"],["25/2","20","27/4","1/4","1"],["0","25/4","35/4","1/2","-2"],["0","0","5/4","9/4","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-25/2","-105/4","-67/4","-3","0"],["0","25/2","20","27/4","1/4","1"],["0","0","25/4","35/4","1/2","-2"],["0","0","0","5/4","9/4","1"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["3/2"],["-5/4"],["1/4"]]},"deriv_order":0},{"coeff":{"coeffs":[["15/4","7/2"],["-5/4","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["5/4","9/4","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","25","45","20","0","0"],["0","-25","-45","-30","-18","-8"],["0","0","0","10","18","8"]]},"deriv_order":0}],"row_multiplier":["5/4","9/4","1"],"start_base":2,"step_multiplier":{"coeffs":[["25/16","45/8","121/16","9/2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["-25/2","-115/2","-211/2","-193/2","-44","-8"],["75/16","20","543/16","229/8","12","2"]]},"deriv_order":0},{"coeff":{"coeffs":[["-125/16","-675/16","-1515/16","-1809/16","-303/4","-27","-4"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["4","4"],["-3","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["5","9","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-1"],["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 2) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = f(n, theta) = 1 + (1) theta; so the candidate rows satisfy the input rule at every base n >= 2 (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":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 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":"f(n, theta) = 1 + (1) theta","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-7","-6"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (1) theta, for every n >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"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 multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n"},"verdict":"open"},"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 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":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 3, with a_n(x) = ((3 n - 7)/(n - 2)) + ((2 n - 6)/(n - 2)) x and b_n(x) = ((-5)/(n - 2)) + ((-9)/(n - 2)) x + ((-4)/(n - 2)) x^2; open hypotheses: earlier_seeds: P_0 is zero or not real-rooted with every root in (-inf, 0) (exact); degree_step_positive: fails at n = 3: lambda_n = (-2 n + 6)/(n - 2) = 0 there (exact); derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = -5 is not the zero polynomial; 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 = 1 (not a disproof): fails at n = 3: [x^1] b_n = -9 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 3, with A_n(x) = ((3 n - 7)/(n - 2)) + ((2 n - 6)/(n - 2)) x, V_n(x) = ((-5)/(n - 2)) + ((-9)/(n - 2)) x + ((-4)/(n - 2)) x^2, W_n(x) = 0; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); seed_degree_step: a seed row of the induction is zero; seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); degree_step_positive: fails at n = 3: lambda_n = (-2 n + 6)/(n - 2) = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_1(n, x) = -5 is not the zero polynomial; derivative_coefficient_nonpositive: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 1 (not a disproof): fails at n = 3: [x^1] V_n = -9 there (exact); derivative_or_lag_coefficient_negative: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 1 (not a disproof): fails at n = 3: [x^1] V_n = -9 there (exact)","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"HypothesisFailed: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 3, with L = -5/4, U = -1, alpha_n = (2 n - 6)/(n - 2), beta_n = (3 n - 7)/(n - 2), kappa_n = (-4)/(n - 2); open hypotheses: earlier_seeds: P_0 is zero, not real-rooted, or has a root outside [-5/4, -1] (exact); slope_positive: fails at n = 3: a_n = -1/2 n + 3/2 = 0 there (exact); upper_endpoint_threshold: no hit before n = 3; E(3) = e(3) + 0 = -2 < 0 with no hit in between (exact); when a_n > 0 for every n <= 3, the criterion puts a root of P_3 outside the interval","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":"HypothesisFailed: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 3 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 3, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 6 - 2 n + 20 x - 6 n x + 15 x^2 - 5 n x^2, B = 4 x + 9 x^2 + 5 x^3, C = 0) / (n - 2); open hypotheses: degree_step: delta = 2, deg P_(n-1) = 2 n - 6: fails at n = 3: Lambda(n, deg P_(n-1)) = 5 n - 15 = 0 there (exact); weyl_step_interlacing: at n = 3: deg A = 1, deg B = 3: neither deg B = deg A nor deg B = deg A + 1; weyl_step_leading_coefficients_one_signed: fails at n = 4: A_a B_(a+1) = -25 n + 75 = -25 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (1) theta for every n >= 0 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}