{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-12","-18"],["-9","-18"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["5","27","36"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["6"],["3"]]},"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":[["3"],["3"]]},"terms":[{"coeff":{"coeffs":[["3","0"],["27","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-68","-99","-36"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["2","1"],"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"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":[["-12","-18"],["-9","-18"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["5","27","36"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["6"],["3"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["6"],["3"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["180","972","1296"],["180","972","1296"],["45","243","324"]]},"step_quotients":[[{"coeff":{"coeffs":[["-360","-3024","-8424","-7776"],["-360","-3024","-8424","-7776"],["-90","-756","-2106","-1944"]]},"deriv_order":0},{"coeff":{"coeffs":[["150","1620","6534","11664","7776"],["75","810","3267","5832","3888"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-150","-1620","-6534","-11664","-7776"],["-75","-810","-3267","-5832","-3888"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["9"],["9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["3","18"],["-15","-36"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["5","27","36"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["5400","58320","235224","419904","279936"],["8100","87480","352836","629856","419904"],["4050","43740","176418","314928","209952"],["675","7290","29403","52488","34992"]]},"step_quotients":[[{"coeff":{"coeffs":[["16200","174960","705672","1259712","839808"],["40500","437400","1764180","3149280","2099520"],["36450","393660","1587762","2834352","1889568"],["14175","153090","617463","1102248","734832"],["2025","21870","88209","157464","104976"]]},"deriv_order":0},{"coeff":{"coeffs":[["-10800","-132840","-645408","-1545480","-1819584","-839808"],["-24300","-311040","-1583388","-4006584","-5038848","-2519424"],["-16200","-211410","-1099332","-2847474","-3674160","-1889568"],["-3375","-44550","-234495","-615276","-804816","-419904"]]},"deriv_order":1},{"coeff":{"coeffs":[["4500","72900","490860","1758348","3534192","3779136","1679616"],["4500","72900","490860","1758348","3534192","3779136","1679616"],["1125","18225","122715","439587","883548","944784","419904"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","-16200","-174960","-705672","-1259712","-839808"],["8100","87480","352836","629856","419904","0"],["8100","99630","484056","1159110","1364688","629856"],["2025","25920","131949","333882","419904","209952"]]},"deriv_order":0},{"coeff":{"coeffs":[["-4500","-72900","-490860","-1758348","-3534192","-3779136","-1679616"],["-4500","-72900","-490860","-1758348","-3534192","-3779136","-1679616"],["-1125","-18225","-122715","-439587","-883548","-944784","-419904"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["3"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1"]]},"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 = P((1) x + (1)); 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":["Polynomial.Splits.comp_of_degree_le_one"],"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":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (1))","kind":"preserver_image","operator":{"intercept":"1","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)), 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 (-inf, -1]","input_root_region":{"upper":{"closed":true,"value":"-1"}},"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), V_n(x) = 0, W_n(x) = (1) + (3) x; open hypotheses: degree_step_positive: fails at n = 2: lambda_n = 0 = 0 there (exact); 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), B_n(x) = (1) + (3) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/3] (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), B_n(x) = (1) + (3) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/3] (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 = P((1) x + (1)) 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, -1]"},"kind":"preserver_image"}