{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","8","16"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"],["6","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-9","-12"],["-7","-12"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1","6","8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3"],["1"]]},"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"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","0"],["19","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-15","-22","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["3","1"],"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","8","16"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"],["6","8"]]},"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":[["-9","-12"],["-7","-12"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1","6","8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["3"],["1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["9","54","72"],["6","36","48"],["1","6","8"]]},"step_quotients":[[{"coeff":{"coeffs":[["-12","-120","-384","-384"],["-19","-190","-608","-608"],["-8","-80","-256","-256"],["-1","-10","-32","-32"]]},"deriv_order":0},{"coeff":{"coeffs":[["12","144","624","1152","768"],["7","84","364","672","448"],["1","12","52","96","64"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-12","-144","-624","-1152","-768"],["-7","-84","-364","-672","-448"],["-1","-12","-52","-96","-64"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["8"],["8"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-4","0"],["-8","-16"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1","6","8"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["27","324","1404","2592","1728"],["27","324","1404","2592","1728"],["9","108","468","864","576"],["1","12","52","96","64"]]},"step_quotients":[[{"coeff":{"coeffs":[["576","6912","29952","55296","36864"],["1392","16704","72384","133632","89088"],["1240","14880","64480","119040","79360"],["520","6240","27040","49920","33280"],["104","1248","5408","9984","6656"],["8","96","416","768","512"]]},"deriv_order":0},{"coeff":{"coeffs":[["-432","-5760","-29376","-71424","-82944","-36864"],["-684","-9312","-48816","-123072","-149760","-70656"],["-384","-5296","-28224","-72640","-90624","-44032"],["-92","-1280","-6896","-17984","-22784","-11264"],["-8","-112","-608","-1600","-2048","-1024"]]},"deriv_order":1},{"coeff":{"coeffs":[["36","648","4752","18144","38016","41472","18432"],["33","594","4356","16632","34848","38016","16896"],["10","180","1320","5040","10560","11520","5120"],["1","18","132","504","1056","1152","512"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["396","5184","25776","60480","66816","27648"],["615","8208","41916","102096","118848","52992"],["341","4608","23924","59568","71360","33024"],["81","1104","5796","14640","17856","8448"],["7","96","508","1296","1600","768"]]},"deriv_order":0},{"coeff":{"coeffs":[["-36","-648","-4752","-18144","-38016","-41472","-18432"],["-33","-594","-4356","-16632","-34848","-38016","-16896"],["-10","-180","-1320","-5040","-10560","-11520","-5120"],["-1","-18","-132","-504","-1056","-1152","-512"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","8","16"],["1","8","16"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-15","-20"],["-6","-8"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["4"],["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"],["3","4"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","8","16"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"],["6","8"]]},"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 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":"Unresolved: P_n = (a_n + b_n (x + 1/4)) P_(n-1) - c_n (x + 1/4)^2 P_(n-2) for n >= 2, with a_n = (4 n + 2)/(n + 2), b_n = (8 n + 4)/(n + 2), c_n = (16 n - 16)/(n + 2); Q_n(y) = y^(d_n) P_n(1/y - 1/4), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: root_region: the region contains the pole x = -1/4 of the coordinate change y = 1/(x + 1/4)","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) = ((6 n + 3)/(n + 2)) + ((8 n + 4)/(n + 2)) x, V_n(x) = 0, W_n(x) = ((-n + 1)/(n + 2)) + ((-8 n + 8)/(n + 2)) x + ((-16 n + 16)/(n + 2)) x^2; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = -n + 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 = 1 (not a disproof): fails at n = 2: [x^1] W_n = -2 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 = 1 (not a disproof): fails at n = 2: [x^1] W_n = -2 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) = ((6 n + 3)/(n + 2)) + ((8 n + 4)/(n + 2)) x, B_n(x) = ((-n + 1)/(n + 2)) + ((-8 n + 8)/(n + 2)) x + ((-16 n + 16)/(n + 2)) x^2: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (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) = ((6 n + 3)/(n + 2)) + ((8 n + 4)/(n + 2)) x, B_n(x) = ((-n + 1)/(n + 2)) + ((-8 n + 8)/(n + 2)) x + ((-16 n + 16)/(n + 2)) x^2","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (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; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)"},"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"}