{"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":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","0"],["2","0"],["0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"],["2","0"],["-1","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"veronese_section_2_1 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles."},"identifying_rows":4,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["1","-2","1"],["4","-4","0"],["6","-2","0"],["4","0","0"],["1","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-15","6"],["-28","4"],["-18","0"],["-4","0"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["25","-2"],["24","0"],["6","0"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["-10"],["-4"]]},"deriv_order":1,"shift":3},{"coeff":{"coeffs":[["1"]]},"deriv_order":1,"shift":4}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2,3],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["0"],["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-81","18","-1"],["108","-12","0"],["-54","2","0"],["12","0","0"],["-1","0","0"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-65","10"],["76","-4"],["-30","0"],["4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-25","2"],["24","0"],["-6","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-6"],["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","0"],["2","0"],["0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"],["2","0"],["-1","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["1","-1"],["2","0"],["1","0"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["-3"],["-2"]]},"deriv_order":1}],"shift":1},{"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":1}],"shift":2}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","-1"],["2","0"],["1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3"],["-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"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 = 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 4 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":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","0"],["2","0"],["0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"],["2","0"],["-1","0"]]},"deriv_order":0,"offset":2,"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":{"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 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":"open"},"notes":[],"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","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: coefficients must be index-independent; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) is not of the form gamma(n) w(x)","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":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (2 n - 1), V_n(x) = 0, W_n(x) = (-n^2 + 2 n - 1) + (1) x; open hypotheses: seed_roots_negative: P_0 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 = 2: lambda_n = 0 = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = -n^2 + 2 n - 1 is not the zero polynomial","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":"HypothesisFailed: P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 2; open hypotheses: seed_rows: P_0 is zero or not a polynomial","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"no lean_proved catalog entry equals the parent; 5 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A073424:lean_proved, oeis_catalog:A129184:lean_proved, oeis_catalog:A132440:lean_proved, oeis_catalog:A216973:lean_proved, oeis_catalog:A395497:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"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 = phi(n, D) = (1) D 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"}