{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(k,n-k)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-8","16","0","-16","8"],["4","-12","12","-4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-5","3","17","-19","0","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["4","-12","12","-4"],["2","-6","6","-2"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","4","2"],["2","4","2"]]},"terms":[{"coeff":{"coeffs":[["17","76","122","84","21"],["-9","-44","-82","-68","-21"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","37","102","134","85","21"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["2","1"],"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-8","-8"],["4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-5","-12","-4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["4"],["2"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["4","-12","12","-4"],["2","-6","6","-2"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["80","352","448","128"],["80","352","448","128"],["20","88","112","32"]]},"step_quotients":[[{"coeff":{"coeffs":[["40","256","576","512","128"],["140","736","1312","896","192"],["60","304","512","320","64"]]},"deriv_order":0},{"coeff":{"coeffs":[["-100","-680","-1696","-1856","-832","-128"],["-50","-340","-848","-928","-416","-64"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-100","-480","-736","-384","-64"],["-50","-240","-368","-192","-32"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-12","-16"],["4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["18","48","24"],["-10","-24","-8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["5","22","28","8"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["1600","14080","48896","83968","72704","28672","4096"],["2400","21120","73344","125952","109056","43008","6144"],["1200","10560","36672","62976","54528","21504","3072"],["200","1760","6112","10496","9088","3584","512"]]},"step_quotients":[[{"coeff":{"coeffs":[["-3200","-34560","-154112","-363520","-481280","-348160","-122880","-16384"],["-4800","-55040","-259328","-643072","-889856","-667648","-241664","-32768"],["-800","-15040","-94848","-286464","-456192","-377856","-145408","-20480"],["1200","8960","22592","14080","-29440","-51200","-25600","-4096"],["400","3520","12224","20992","18176","7168","1024","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["3200","37760","188672","517632","844800","829440","471040","139264","16384"],["-800","-7040","-21248","-13824","61440","153600","143360","57344","8192"],["-3200","-35360","-162752","-402048","-572160","-468480","-209920","-47104","-4096"],["-1000","-11200","-52480","-132864","-195840","-168960","-81920","-20480","-2048"]]},"deriv_order":1},{"coeff":{"coeffs":[["2000","26400","149760","475648","923136","1121280","839680","368640","86016","8192"],["2000","26400","149760","475648","923136","1121280","839680","368640","86016","8192"],["500","6600","37440","118912","230784","280320","209920","92160","21504","2048"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","3200","28160","97792","167936","145408","57344","8192"],["-1600","-10880","-20736","13824","95232","116736","53248","8192"],["-1600","-13280","-41856","-59520","-30720","7680","10240","2048"],["-400","-3520","-12224","-20992","-18176","-7168","-1024","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["2000","22400","104960","265728","391680","337920","163840","40960","4096"],["2000","22400","104960","265728","391680","337920","163840","40960","4096"],["500","5600","26240","66432","97920","84480","40960","10240","1024"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["2"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), 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 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":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.splits_fPolynomial"],"lean_route":"fTransform","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":"((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0)","kind":"preserver_image","operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0), for every n >= 0"},"parent_proof":{"route":"liu_wang_three_term","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 [-1, 0]","input_root_region":{"lower":{"closed":true,"value":"-1"},"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":["the division remainder is removed by row annihilators of the shifted parent"],"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: support must satisfy offset = 1 + ell * t_degree; fixed_delay_predecessor_window: only P_(n-1) and one term w(n) t P_(n-L) are supported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","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) = (1) + (2) x, V_n(x) = 0, W_n(x) = (1) + (2) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = 1 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 1 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 1 there (exact)","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":"P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 2, with A_n(x) = (1) + (2) x, B_n(x) = (1) + (2) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/2] (so its coefficients have one sign); for n >= 1, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 2","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":1,"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-2) for n >= 2, with A_n(x) = (1) + (2) x, B_n(x) = (1) + (2) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/2] (so its coefficients have one sign); for n >= 1, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 2"],"theorem":"liu_wang_three_term: L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"liu_wang_three_term","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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), 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 [-1, 0]"},"kind":"preserver_image"}