{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["11","14","4"],["41","76","44","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A007318 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-396","-2181","361","3913","1308","-384"],["-144","-1122","1700","4819","952","-704"],["0","-126","888","1957","-53","-504"],["0","0","120","298","-192","-176"],["0","0","0","12","-51","-30"],["0","0","0","0","-4","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-216","-2268","-3879","9763","16651","3172","-1664"],["0","-432","-342","11868","16809","696","-2880"],["0","0","126","3792","5399","-1671","-1928"],["0","0","0","360","606","-960","-624"],["0","0","0","0","12","-185","-98"],["0","0","0","0","0","-12","-6"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-270","-1746","1149","11211","10053","368","-1024"],["0","0","-144","2658","10160","7759","-1748","-1664"],["0","0","0","504","2298","1367","-1943","-1024"],["0","0","0","0","120","-62","-672","-296"],["0","0","0","0","0","-18","-91","-40"],["0","0","0","0","0","0","-4","-2"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","-54","-270","666","2170","1228","-188","-128"],["0","0","0","0","756","1692","588","-540","-192"],["0","0","0","0","126","282","-118","-378","-104"],["0","0","0","0","0","0","-72","-96","-24"],["0","0","0","0","0","0","-6","-8","-2"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["180","255","-835","-595","180"],["72","102","-1084","-713","372"],["0","0","-450","-285","305"],["0","0","-60","-38","124"],["0","0","0","0","25"],["0","0","0","0","2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","540","765","-2505","-1785","540"],["0","216","306","-3252","-2139","1116"],["0","0","0","-1350","-855","915"],["0","0","0","-180","-114","372"],["0","0","0","0","0","75"],["0","0","0","0","0","6"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","180","255","-835","-595","180"],["0","0","72","102","-1084","-713","372"],["0","0","0","0","-450","-285","305"],["0","0","0","0","-60","-38","124"],["0","0","0","0","0","0","25"],["0","0","0","0","0","0","2"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["11","56","36"]],"recurrence":{"denominator":{"coeffs":[["108","153","-141","-129","12"],["72","102","-364","-257","44"],["0","0","-270","-171","63"],["0","0","-60","-38","44"],["0","0","0","0","15"],["0","0","0","0","2"]]},"terms":[{"coeff":{"coeffs":[["252","1185","571","-765","-448","36"],["144","870","-284","-1751","-666","114"],["0","126","-528","-1135","-257","136"],["0","0","-120","-250","28","76"],["0","0","0","-12","31","20"],["0","0","0","0","4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["54","270","-36","-760","-588","-62","18"],["0","0","-504","-1128","-632","40","48"],["0","0","-126","-282","-62","138","44"],["0","0","0","0","48","64","16"],["0","0","0","0","6","8","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["125/8","75/4","15/2","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["11","14","4"],["41","76","44","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","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":[["-11","-6"],["-4","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-6","-8","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["5"],["2"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["40500","165375","64125","-458625","-720250","-379875","-40875","21125","4500"],["48600","198450","-91800","-1107225","-1494300","-687100","2200","73475","12900"],["19440","79380","-205470","-999765","-1199595","-431090","120880","102515","15785"],["2592","10584","-117396","-430302","-454696","-70812","144284","76002","10688"],["0","0","-27000","-89100","-72675","38000","76950","32700","4325"],["0","0","-2160","-7128","486","19840","21556","8216","1046"],["0","0","0","0","1260","3360","3080","1120","140"],["0","0","0","0","72","192","176","64","8"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","121500","496125","192375","-1375875","-2160750","-1139625","-122625","63375","13500"],["0","145800","595350","-275400","-3321675","-4482900","-2061300","6600","220425","38700"],["0","58320","238140","-616410","-2999295","-3598785","-1293270","362640","307545","47355"],["0","7776","31752","-352188","-1290906","-1364088","-212436","432852","228006","32064"],["0","0","0","-81000","-267300","-218025","114000","230850","98100","12975"],["0","0","0","-6480","-21384","1458","59520","64668","24648","3138"],["0","0","0","0","0","3780","10080","9240","3360","420"],["0","0","0","0","0","216","576","528","192","24"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","40500","165375","64125","-458625","-720250","-379875","-40875","21125","4500"],["0","0","48600","198450","-91800","-1107225","-1494300","-687100","2200","73475","12900"],["0","0","19440","79380","-205470","-999765","-1199595","-431090","120880","102515","15785"],["0","0","2592","10584","-117396","-430302","-454696","-70812","144284","76002","10688"],["0","0","0","0","-27000","-89100","-72675","38000","76950","32700","4325"],["0","0","0","0","-2160","-7128","486","19840","21556","8216","1046"],["0","0","0","0","0","0","1260","3360","3080","1120","140"],["0","0","0","0","0","0","72","192","176","64","8"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["75","100","25"],["60","80","20"],["12","16","4"]]},"step_quotients":[[{"coeff":{"coeffs":[["195","350","185","30"],["138","250","134","22"],["24","44","24","4"]]},"deriv_order":0},{"coeff":{"coeffs":[["-90","-240","-220","-80","-10"],["-36","-96","-88","-32","-4"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-180","-480","-440","-160","-20"],["-72","-192","-176","-64","-8"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["-6"],["-5"],["-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["3","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3","4","1"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","141750","425250","735750","1159500","1187250","527250","1250","-56000","-10000"],["0","170100","510300","970650","1722150","1780200","635200","-188750","-160450","-25000"],["0","68040","204120","444960","917460","913980","112080","-350200","-182780","-25900"],["0","9072","27216","73008","189768","140874","-155856","-254740","-107144","-14230"],["0","0","0","-864","3888","-24948","-90560","-90776","-34288","-4372"],["0","0","0","-864","-2304","-8520","-17856","-15760","-5696","-712"],["0","0","0","0","0","-432","-1152","-1056","-384","-48"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","60750","202500","236250","195750","188250","55000","-70250","-45250","-7000"],["0","0","72900","243000","229500","133650","126900","-93500","-250300","-121550","-17400"],["0","0","29160","97200","28350","-81540","-111690","-273000","-317670","-132420","-17910"],["0","0","3888","12960","-35100","-100872","-139872","-214640","-196876","-75176","-9768"],["0","0","0","0","-13176","-32400","-51696","-76448","-64504","-23504","-2976"],["0","0","0","0","-1296","-3456","-7488","-12672","-10704","-3840","-480"],["0","0","0","0","0","0","-288","-768","-704","-256","-32"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["1125","3000","2750","1000","125"],["1350","3600","3300","1200","150"],["540","1440","1320","480","60"],["72","192","176","64","8"]]},"step_quotients":[[{"coeff":{"coeffs":[["-2700","-7200","-6600","-2400","-300"],["-3735","-9960","-9130","-3320","-415"],["-1917","-5112","-4686","-1704","-213"],["-432","-1152","-1056","-384","-48"],["-36","-96","-88","-32","-4"]]},"deriv_order":0},{"coeff":{"coeffs":[["675","2250","2850","1700","475","50"],["540","1800","2280","1360","380","40"],["108","360","456","272","76","8"]]},"deriv_order":1},{"coeff":{"coeffs":[["675","2700","4275","3400","1425","300","25"],["540","2160","3420","2720","1140","240","20"],["108","432","684","544","228","48","4"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["4275","13650","16450","9300","2475","250"],["4320","13770","16560","9340","2480","250"],["1404","4464","5352","3008","796","80"],["144","456","544","304","80","8"]]},"deriv_order":0},{"coeff":{"coeffs":[["1350","5400","8550","6800","2850","600","50"],["1080","4320","6840","5440","2280","480","40"],["216","864","1368","1088","456","96","8"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["72","192","176","64","8"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-4","-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 = f(n, theta) = 1 + (2) theta + (1) theta^2; 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":{"from_index":1,"justification":"the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = f(n, theta) = 1 + (2) theta + (1) theta^2 is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 1","kind":"weak","orientation":"unordered","status":"claimed"},"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 2, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 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":{"interlacing":{"from_index":1,"justification":{"premises":["the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 1; T = f(n, theta) = 1 + (2) theta + (1) theta^2 is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 1"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"proved"},"lean_replay":"not_attempted","operation":{"description":"f(n, theta) = 1 + (2) theta + (1) theta^2","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["11","14","4"],["41","76","44","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2, for every n >= 0"},"parent_proof":{"route":"three_term_recurrence","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":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":0,"interlacing":"strict","interlacing_from":1,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"},"hypotheses":[{"evidence":"f(n) a_n = 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 2 at n = 2 + u have one sign)","name":"leading_coefficient_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2"]},"numerator":["2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"Q_0 has degree 2, Q_1 degree 3; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_0 and Q_1 have simple real roots, degrees e and e + 1, and Q_0 strictly interlaces Q_1","status":{"kind":"verified"}},{"evidence":"beta_2 = 1/4 (exact)","name":"first_coefficient_positive","statement":"beta_2 = c_2 L_0 / (a_2 L_1) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = 1/4 > 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of beta_n = 1/4 at n = 3 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1/4"]},"numerator":["1/4"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"seeds checked exactly; positive-ratio induction at t = 0 with g(n) = 1 ((t - alpha_n)/2): g(n) > 0 for n >= 2, r_2 >= g(2) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 3 + u have one sign)","name":"root_upper_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["3/4"]},"numerator":["3/4"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"every root of P_n is < 0 for every n >= 0 (region (-inf, 0) for P)","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n x + b_n) P_(n-1) - c_n P_(n-2) for n >= 2, with a_n = 2, b_n = 4, c_n = 1","recurrence":{"first_index":0,"initial_polynomials":[["11","14","4"],["41","76","44","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"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":"P_n = (a_n x + b_n) P_(n-1) - c_n P_(n-2) for n >= 2, with a_n = 2, b_n = 4, c_n = 1: every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)","route":"three_term_recurrence","success":true,"variant":"three_term_recurrence"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (a_n x + b_n) P_(n-1) - c_n P_(n-2) for n >= 2, with a_n = 2, b_n = 4, c_n = 1","every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"],"theorem":"three_term_recurrence: three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"three_term_recurrence","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 = f(n, theta) = 1 + (2) theta + (1) theta^2 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"}