{"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":[["1"],["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-2"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k)."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["-300","-1180","-1708","-1212","-448","-72"],["0","-550","-1540","-1556","-690","-120"],["0","0","-300","-571","-355","-74"],["0","0","0","-55","-68","-20"],["0","0","0","0","-3","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["500","3300","8740","11996","9048","3568","576"],["0","600","3660","8296","8580","4112","744"],["0","0","-450","-520","524","814","232"],["0","0","0","-600","-973","-465","-62"],["0","0","0","0","-145","-168","-44"],["0","0","0","0","0","-9","-6"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","625","4125","10925","14995","11310","4460","720"],["0","0","1575","7820","15067","14058","6372","1128"],["0","0","0","950","3585","4934","2915","620"],["0","0","0","0","75","354","395","126"],["0","0","0","0","0","-30","-23","0"],["0","0","0","0","0","0","-3","-2"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","125","825","2185","2999","2262","892","144"],["0","0","0","375","1800","3355","3054","1364","240"],["0","0","0","0","300","1025","1298","721","148"],["0","0","0","0","0","75","185","150","40"],["0","0","0","0","0","0","5","9","4"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["100","185","101","18","4"],["0","125","180","82","12"],["0","0","75","67","13"],["0","0","0","15","6"],["0","0","0","0","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","300","555","303","54","12"],["0","0","375","540","246","36"],["0","0","0","225","201","39"],["0","0","0","0","45","18"],["0","0","0","0","0","3"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","100","185","101","18","4"],["0","0","0","125","180","82","12"],["0","0","0","0","75","67","13"],["0","0","0","0","0","15","6"],["0","0","0","0","0","0","1"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["-100","-60","4","12","0"],["100","-65","-34","-5","0"],["0","125","-45","-29","-1"],["0","0","75","7","-1"],["0","0","0","15","1"],["0","0","0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-300","-630","-468","-172","-48","-8"],["300","80","-472","-407","-124","-16"],["0","550","640","173","3","-2"],["0","0","300","351","113","14"],["0","0","0","55","53","10"],["0","0","0","0","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-125","-450","-685","-594","-326","-108","-16"],["0","-375","-1200","-1530","-993","-336","-48"],["0","0","-300","-800","-773","-325","-52"],["0","0","0","-75","-165","-114","-24"],["0","0","0","0","-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","0","0","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-2"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-3","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["5","9","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0"],["2500","13625","31275","39125","28861","12770","3396","576","64"],["0","3125","15750","33375","38460","25962","10236","2176","192"],["0","0","1875","8425","15430","14677","7597","2008","208"],["0","0","0","375","1500","2355","1806","672","96"],["0","0","0","0","25","90","121","72","16"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","7500","40875","93825","117375","86583","38310","10188","1728","192"],["0","0","9375","47250","100125","115380","77886","30708","6528","576"],["0","0","0","5625","25275","46290","44031","22791","6024","624"],["0","0","0","0","1125","4500","7065","5418","2016","288"],["0","0","0","0","0","75","270","363","216","48"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","2500","13625","31275","39125","28861","12770","3396","576","64"],["0","0","0","3125","15750","33375","38460","25962","10236","2176","192"],["0","0","0","0","1875","8425","15430","14677","7597","2008","208"],["0","0","0","0","0","375","1500","2355","1806","672","96"],["0","0","0","0","0","0","25","90","121","72","16"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","0","0"],["0","0","0"],["5","9","4"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0"],["0","0","0","0"],["-10","-28","-26","-8"],["-10","-28","-26","-8"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["25","90","121","72","16"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["-25","-90","-121","-72","-16"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-1"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["7","6"],["-5","-4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["5","9","4"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","-4375","-17625","-19175","17125","61430","62932","31464","7552","640"],["0","0","-1875","-1250","17975","48920","55490","32144","9248","1024"],["0","0","0","-250","850","5940","10810","8994","3568","544"],["0","0","0","0","0","150","540","726","432","96"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","-1875","-8375","-12425","-1165","18180","24044","14400","4288","512"],["0","0","0","-1250","-3375","1100","14205","21764","15456","5440","768"],["0","0","0","0","-375","-600","1485","4710","4824","2208","384"],["0","0","0","0","0","0","100","360","484","288","64"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["25","90","121","72","16"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["25","90","121","72","16"],["25","90","121","72","16"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","0"],["50","230","422","386","176","32"],["-125","-550","-965","-844","-368","-64"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["125","675","1515","1809","1212","432","64"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","0"],["-100","-460","-844","-772","-352","-64"],["75","320","543","458","192","32"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["-125","-675","-1515","-1809","-1212","-432","-64"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["25","90","121","72","16"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0"],["-1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0"],["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 1) 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 >= 1 (the exceptional bases checked exactly), and agreement on the 2 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":["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":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_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":[["1"],["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-2"],["1","1"]]},"deriv_order":0,"offset":2,"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":"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":"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_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":"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 >= 3, with A_n(x) = ((n - 2)/(n - 2)), V_n(x) = 0, W_n(x) = ((n - 2)/(n - 2)) + ((n - 2)/(n - 2)) x; open hypotheses: degree_step_positive: fails at n = 3: lambda_n = (0)/(n - 2) = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = n - 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 = 3: [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 = 3: [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 >= 3, with A_n(x) = ((n - 2)/(n - 2)), B_n(x) = ((n - 2)/(n - 2)) + ((n - 2)/(n - 2)) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1] (so its coefficients have one sign); for n >= 2, 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 >= 3","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":2,"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 >= 3, with A_n(x) = ((n - 2)/(n - 2)), B_n(x) = ((n - 2)/(n - 2)) + ((n - 2)/(n - 2)) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1] (so its coefficients have one sign); for n >= 2, 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 >= 3"],"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 = 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"}