{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013609 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ..."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-4","-4"],["1","4"],["-3","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-9","-24","-12"],["6","12","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-3","-12","-14","-4"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["1"],["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":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","9","6"],["4","-1","-15","-9"],["0","5","6","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","20","33","12"],["0","-4","-16","-18","-6"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","4","14","12","3"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1","1"],"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","4"]]},"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":[["-4","-4"],["1","4"],["-3","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-9","-24","-12"],["6","12","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-3","-12","-14","-4"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","0","0","27","324","1674","4860","8676","9792","6920","2928","672","64"],["0","0","0","81","972","5022","14580","26028","29376","20760","8784","2016","192"],["0","0","0","81","972","5022","14580","26028","29376","20760","8784","2016","192"],["0","0","0","27","324","1674","4860","8676","9792","6920","2928","672","64"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","54","864","5940","23112","56232","88992","92176","61216","24768","5504","512"],["0","0","0","81","1620","12798","54756","142668","237600","255768","174864","72288","16320","1536"],["0","0","0","-81","-324","2754","25596","90612","178848","214248","157296","68256","15936","1536"],["0","0","0","-189","-2052","-9126","-20628","-21852","864","29896","34864","18720","4928","512"],["0","0","0","-81","-972","-5022","-14580","-26028","-29376","-20760","-8784","-2016","-192","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-81","-1296","-9126","-37260","-97740","-172368","-207672","-170160","-92512","-31680","-6144","-512"],["0","0","0","0","-324","-4212","-23976","-78408","-162432","-221616","-200544","-118176","-43200","-8832","-768"],["0","0","0","243","3240","18954","63828","136404","192240","179784","109392","41184","8640","768","0"],["0","0","0","162","2268","14040","50544","117072","182304","193728","139776","66848","20160","3456","256"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","-81","-1296","-9288","-39312","-109080","-208224","-279168","-263616","-173200","-76928","-21888","-3584","-256"],["0","0","0","-162","-2592","-18576","-78624","-218160","-416448","-558336","-527232","-346400","-153856","-43776","-7168","-512"],["0","0","0","-81","-1296","-9288","-39312","-109080","-208224","-279168","-263616","-173200","-76928","-21888","-3584","-256"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","-81","-1134","-6858","-23328","-48492","-61992","-44808","-11136","8096","7488","2304","256"],["0","-243","-3483","-21546","-75006","-160056","-212004","-163800","-54168","15504","20448","6720","768"],["0","-243","-3645","-23490","-85050","-189216","-264060","-222552","-95688","-2064","16416","6336","768"],["0","-81","-1377","-9774","-38394","-92232","-140076","-132936","-73416","-18256","1440","1728","256"],["0","0","-81","-972","-5022","-14580","-26028","-29376","-20760","-8784","-2016","-192","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","81","1215","7992","30186","71820","110484","106800","55944","3040","-15584","-9792","-2560","-256"],["0","162","2592","18252","74412","194184","338040","395904","305616","145856","35680","576","-1664","-256"],["0","81","1539","12528","58266","172908","344628","471408","443400","282592","118112","30528","4352","256"],["0","0","162","2268","14040","50544","117072","182304","193728","139776","66848","20160","3456","256"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","-81","-1296","-9288","-39312","-109080","-208224","-279168","-263616","-173200","-76928","-21888","-3584","-256"],["0","0","-162","-2592","-18576","-78624","-218160","-416448","-558336","-527232","-346400","-153856","-43776","-7168","-512"],["0","0","-81","-1296","-9288","-39312","-109080","-208224","-279168","-263616","-173200","-76928","-21888","-3584","-256"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["3","3","-4"],["0","3","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","-3","10","8"],["0","-6","-12","-4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","3","12","14","4"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["0","0","0","27","324","1674","4860","8676","9792","6920","2928","672","64"],["0","0","0","81","972","5022","14580","26028","29376","20760","8784","2016","192"],["0","0","0","81","972","5022","14580","26028","29376","20760","8784","2016","192"],["0","0","0","27","324","1674","4860","8676","9792","6920","2928","672","64"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","81","1134","6858","23328","48492","61992","44808","11136","-8096","-7488","-2304","-256"],["0","0","0","243","3483","21546","75006","160056","212004","163800","54168","-15504","-20448","-6720","-768"],["0","0","0","243","3645","23490","85050","189216","264060","222552","95688","2064","-16416","-6336","-768"],["0","0","0","81","1377","9774","38394","92232","140076","132936","73416","18256","-1440","-1728","-256"],["0","0","0","0","81","972","5022","14580","26028","29376","20760","8784","2016","192","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-81","-1215","-7992","-30186","-71820","-110484","-106800","-55944","-3040","15584","9792","2560","256"],["0","0","0","-162","-2592","-18252","-74412","-194184","-338040","-395904","-305616","-145856","-35680","-576","1664","256"],["0","0","0","-81","-1539","-12528","-58266","-172908","-344628","-471408","-443400","-282592","-118112","-30528","-4352","-256"],["0","0","0","0","-162","-2268","-14040","-50544","-117072","-182304","-193728","-139776","-66848","-20160","-3456","-256"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","81","1296","9288","39312","109080","208224","279168","263616","173200","76928","21888","3584","256"],["0","0","0","0","162","2592","18576","78624","218160","416448","558336","527232","346400","153856","43776","7168","512"],["0","0","0","0","81","1296","9288","39312","109080","208224","279168","263616","173200","76928","21888","3584","256"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","162","2322","14580","52596","120096","180216","178608","114448","45024","9792","896"],["0","0","486","7047","44712","162810","374868","566676","565200","364104","143856","31392","2880"],["0","0","486","7209","46656","172854","404028","618732","623952","405624","161424","35424","3264"],["0","0","162","2565","17496","67662","163836","258300","266736","176728","71376","15840","1472"],["0","0","0","81","972","5022","14580","26028","29376","20760","8784","2016","192"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-162","-2511","-17280","-69498","-180900","-318708","-385968","-319560","-176240","-61344","-12096","-1024","0"],["0","0","-324","-5184","-36828","-153036","-412344","-754488","-954240","-832848","-492256","-189536","-44352","-5504","-256"],["0","0","-162","-2835","-21816","-97578","-281988","-552852","-750576","-707016","-455792","-195040","-52416","-7936","-512"],["0","0","0","-162","-2268","-14040","-50544","-117072","-182304","-193728","-139776","-66848","-20160","-3456","-256"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","81","1296","9288","39312","109080","208224","279168","263616","173200","76928","21888","3584","256"],["0","0","0","162","2592","18576","78624","218160","416448","558336","527232","346400","153856","43776","7168","512"],["0","0","0","81","1296","9288","39312","109080","208224","279168","263616","173200","76928","21888","3584","256"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-2","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-4","-4"]]},"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 = x^(d_n) P(1/x), 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.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"lean_route":"reversed","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":"x^(d_n) P(1/x), d_n = 0 + 1 (n - 0)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = x^(d_n) P(1/x), 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 (-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":"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: coefficient of P_(n-1) must be 1 after normalization; 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) = 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) = (4) + (4) x, V_n(x) = 0, W_n(x) = (2) + (4) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = 2 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 = 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 = 0 (not a disproof): fails at n = 2: [x^0] 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) = (4) + (4) x, B_n(x) = (2) + (4) 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) = (4) + (4) x, B_n(x) = (2) + (4) 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 = x^(d_n) P(1/x), 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 (-inf, 0]"},"kind":"preserver_image"}