{"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"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-3","-4"],["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-14","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-288","-804","-626","-112","-32","0"],["-720","-4290","-8593","-7516","-3040","-576"],["0","-1704","-6628","-8742","-4688","-960"],["0","0","-1208","-3046","-2380","-592"],["0","0","0","-288","-448","-160"],["0","0","0","0","-20","-16"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["864","7344","26460","48674","47248","23072","4608"],["0","1296","9114","27773","40188","26272","5952"],["0","0","-1560","-3964","-130","4400","1856"],["0","0","0","-2472","-5490","-3364","-496"],["0","0","0","0","-768","-1120","-352"],["0","0","0","0","0","-60","-48"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1080","9972","35286","62564","59368","28928","5760"],["0","0","3600","23190","58347","70904","41200","9024"],["0","0","0","2736","13272","23878","18392","4960"],["0","0","0","0","232","1514","2340","1008"],["0","0","0","0","0","-168","-168","0"],["0","0","0","0","0","0","-20","-16"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","216","2052","7218","12638","11896","5792","1152"],["0","0","0","864","5496","13388","15684","8848","1920"],["0","0","0","0","888","3980","6524","4616","1184"],["0","0","0","0","0","288","912","944","320"],["0","0","0","0","0","0","24","56","32"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["144","402","313","56","16"],["144","642","785","330","64"],["0","240","656","496","100"],["0","0","184","270","76"],["0","0","0","48","28"],["0","0","0","0","4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","432","1206","939","168","48"],["0","432","1926","2355","990","192"],["0","0","720","1968","1488","300"],["0","0","0","552","810","228"],["0","0","0","0","144","84"],["0","0","0","0","0","12"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","144","402","313","56","16"],["0","0","144","642","785","330","64"],["0","0","0","240","656","496","100"],["0","0","0","0","184","270","76"],["0","0","0","0","0","48","28"],["0","0","0","0","0","0","4"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0","0"],["144","162","25","-44","0"],["0","240","104","-26","0"],["0","0","184","78","4"],["0","0","0","48","8"],["0","0","0","0","4"]]},"terms":[{"coeff":{"coeffs":[["-432","-1782","-2547","-1420","-272","-64"],["720","882","-1039","-1982","-888","-128"],["0","1704","3004","1332","36","-16"],["0","0","1208","1894","788","112"],["0","0","0","288","348","80"],["0","0","0","0","20","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-216","-1188","-2610","-2942","-1848","-672","-128"],["0","-864","-3720","-6292","-5276","-2224","-384"],["0","0","-888","-3116","-3932","-2120","-416"],["0","0","0","-288","-816","-720","-192"],["0","0","0","0","-24","-56","-32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1","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"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-3","-4"],["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-14","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["81","4833/8","30213/16","12765/4","50161/16","1795","582","112","16"],["81","5913/8","44541/16","45435/8","109997/16","40505/8","2232","554","64"],["0","135","999","3096","20765/4","10123/2","11393/4","846","100"],["0","0","207/2","5079/8","1591","16603/8","5903/4","536","76"],["0","0","0","27","567/4","585/2","1183/4","146","28"],["0","0","0","0","9/4","21/2","73/4","14","4"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","243","14499/8","90639/16","38295/4","150483/16","5385","1746","336","48"],["0","243","17739/8","133623/16","136305/8","329991/16","121515/8","6696","1662","192"],["0","0","405","2997","9288","62295/4","30369/2","34179/4","2538","300"],["0","0","0","621/2","15237/8","4773","49809/8","17709/4","1608","228"],["0","0","0","0","81","1701/4","1755/2","3549/4","438","84"],["0","0","0","0","0","27/4","63/2","219/4","42","12"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","81","4833/8","30213/16","12765/4","50161/16","1795","582","112","16"],["0","0","81","5913/8","44541/16","45435/8","109997/16","40505/8","2232","554","64"],["0","0","0","135","999","3096","20765/4","10123/2","11393/4","846","100"],["0","0","0","0","207/2","5079/8","1591","16603/8","5903/4","536","76"],["0","0","0","0","0","27","567/4","585/2","1183/4","146","28"],["0","0","0","0","0","0","9/4","21/2","73/4","14","4"]]},"deriv_order":2}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["-1/4"],["1/4"]]},"deriv_order":0},{"coeff":{"coeffs":[["9/8","3/2"],["-1","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["3/4","7/4","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","-1134","-6075","-9072","8661","45156","61404","40112","12352","1280"],["0","-1134","-6723","-9720","18831","81792","115598","80672","27232","3328"],["0","0","-648","-756","10674","41076","64554","51604","20480","3136"],["0","0","0","-108","504","4548","10864","11920","6272","1280"],["0","0","0","0","0","108","504","876","672","192"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-486","-2916","-5886","-1368","13176","23440","18240","6912","1024"],["0","0","-486","-3348","-7416","-750","23946","44766","37648","15584","2560"],["0","0","0","-432","-1692","294","11904","25838","25312","12128","2304"],["0","0","0","0","-162","-324","1206","4848","6488","3904","896"],["0","0","0","0","0","0","72","336","584","448","128"]]},"deriv_order":1}],"row_multiplier":["9/16","21/8","73/16","7/2","1"],"start_base":0,"step_multiplier":{"coeffs":[["9/16","21/8","73/16","7/2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["-9/4","-15","-157/4","-101/2","-32","-8"],["45/16","123/8","533/16","143/4","19","4"]]},"deriv_order":0},{"coeff":{"coeffs":[["-27/8","-189/8","-549/8","-847/8","-183/2","-42","-8"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-2","0"],["-5","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["6","14","8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"shift":1}]},"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":{"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"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-3","-4"],["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-14","-8"]]},"deriv_order":1,"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":{"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"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":"open"},"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: the theorem needs at least two seed rows","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = ((5 n - 3)/(n)) + ((4 n - 4)/(n)) x and b_n(x) = ((-6)/(n)) + ((-14)/(n)) x + ((-8)/(n)) x^2; open hypotheses: degree_step_positive: fails at n = 1: lambda_n = (-4 n + 4)/(n) = 0 there (exact); derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = -6 is not the zero polynomial; derivative_coefficient_nonpositive: b_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] b_n <= 0 fails for k = 1 (not a disproof): fails at n = 1: [x^1] b_n = -14 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"not applicable: the theorem needs two seed rows","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"HypothesisFailed: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 1, with L = -1, U = -3/4, alpha_n = (4 n - 4)/(n), beta_n = (5 n - 3)/(n), kappa_n = (-8)/(n); open hypotheses: slope_positive: fails at n = 1: a_n = -1/2 n + 1/2 = 0 there (exact); upper_endpoint_threshold: no hit before n = 1; E(1) = e(1) + 0 = -1 < 0 with no hit in between (exact); when a_n > 0 for every n <= 1, the criterion puts a root of P_1 outside the interval","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"not applicable: the theorem needs two seed rows","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"HypothesisFailed: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 1 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 1, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 4 - 4 n + 11 x - 9 n x + 6 x^2 - 6 n x^2, B = 8 x + 14 x^2 + 6 x^3, C = 0) / (n); open hypotheses: degree_step: delta = 2, deg P_(n-1) = 2 n - 2: fails at n = 1: Lambda(n, deg P_(n-1)) = 6 n - 6 = 0 there (exact); weyl_step_interlacing: at n = 1: deg A = 1, deg B = 3: neither deg B = deg A nor deg B = deg A + 1; weyl_step_leading_coefficients_one_signed: fails at n = 2: A_a B_(a+1) = -36 n + 36 = -36 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"no lean_proved catalog entry equals the parent; 549 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A001263:lean_proved, oeis_catalog:A003982:lean_proved, oeis_catalog:A007318:lean_proved, oeis_catalog:A008299:lean_proved, oeis_catalog:A008303:lean_proved, oeis_catalog:A008306:lean_proved, oeis_catalog:A008459:lean_proved, oeis_catalog:A008517:lean_proved, oeis_catalog:A008518:lean_proved, oeis_catalog:A008556:lean_proved, oeis_catalog:A008955:lean_proved, oeis_catalog:A008957:lean_proved, oeis_catalog:A008970:lean_proved, oeis_catalog:A008971:lean_proved, oeis_catalog:A010054:lean_proved, oeis_catalog:A011973:lean_proved, oeis_catalog:A013609:lean_proved, oeis_catalog:A013610:lean_proved, oeis_catalog:A013611:lean_proved, oeis_catalog:A013613:lean_proved, oeis_catalog:A013620:lean_proved, oeis_catalog:A013622:lean_proved, oeis_catalog:A023531:lean_proved, oeis_catalog:A024462:lean_proved, oeis_catalog:A026374:lean_proved, oeis_catalog:A026386:lean_proved, oeis_catalog:A027465:lean_proved, oeis_catalog:A027466:lean_proved, oeis_catalog:A027467:lean_proved, oeis_catalog:A028246:lean_proved, oeis_catalog:A028262:lean_proved, oeis_catalog:A028313:lean_proved, oeis_catalog:A028338:lean_proved, oeis_catalog:A028421:lean_proved, oeis_catalog:A029600:lean_proved, oeis_catalog:A029618:lean_proved, oeis_catalog:A029653:lean_proved, oeis_catalog:A030528:lean_proved, oeis_catalog:A033282:lean_proved, oeis_catalog:A034839:lean_proved, oeis_catalog:A034867:lean_proved, oeis_catalog:A034870:lean_proved, oeis_catalog:A035342:lean_proved, oeis_catalog:A036969:lean_proved, oeis_catalog:A038207:lean_proved, oeis_catalog:A038208:lean_proved, oeis_catalog:A038220:lean_proved, oeis_catalog:A038221:lean_proved, oeis_catalog:A038231:lean_proved, oeis_catalog:A038243:lean_proved, oeis_catalog:A038255:lean_proved, oeis_catalog:A038259:lean_proved, oeis_catalog:A038271:lean_proved, oeis_catalog:A038277:lean_proved, oeis_catalog:A038279:lean_proved, oeis_catalog:A038291:lean_proved, oeis_catalog:A038303:lean_proved, oeis_catalog:A038322:lean_proved, oeis_catalog:A038323:lean_proved, oeis_catalog:A038324:lean_proved, oeis_catalog:A038325:lean_proved, oeis_catalog:A038326:lean_proved, oeis_catalog:A038327:lean_proved, oeis_catalog:A038335:lean_proved, oeis_catalog:A038763:lean_proved, oeis_catalog:A039683:lean_proved, oeis_catalog:A039692:lean_proved, oeis_catalog:A039756:lean_proved, oeis_catalog:A039991:lean_proved, oeis_catalog:A046741:lean_proved, oeis_catalog:A046802:lean_proved, oeis_catalog:A048176:lean_proved, oeis_catalog:A048993:lean_proved, oeis_catalog:A049020:lean_proved, oeis_catalog:A049061:lean_proved, oeis_catalog:A049374:lean_proved, oeis_catalog:A049444:lean_proved, oeis_catalog:A049458:lean_proved, oeis_catalog:A049459:lean_proved, oeis_catalog:A049460:lean_proved, oeis_catalog:A051338:lean_proved, oeis_catalog:A051339:lean_proved, oeis_catalog:A051379:lean_proved, oeis_catalog:A051380:lean_proved, oeis_catalog:A051523:lean_proved, oeis_catalog:A054142:lean_proved, oeis_catalog:A055151:lean_proved, oeis_catalog:A055356:lean_proved, oeis_catalog:A056241:lean_proved, oeis_catalog:A056939:lean_proved, oeis_catalog:A059022:lean_proved, oeis_catalog:A059023:lean_proved, oeis_catalog:A059340:lean_proved, oeis_catalog:A059344:lean_proved, oeis_catalog:A059364:lean_proved, oeis_catalog:A059374:lean_proved, oeis_catalog:A059473:lean_proved, oeis_catalog:A060523:lean_proved, oeis_catalog:A060821:lean_proved, oeis_catalog:A060920:lean_proved, oeis_catalog:A060923:lean_proved, oeis_catalog:A061896:lean_proved, oeis_catalog:A062190:lean_proved, oeis_catalog:A062196:lean_proved, oeis_catalog:A062264:lean_proved, oeis_catalog:A066325:lean_proved, oeis_catalog:A069466:lean_proved, oeis_catalog:A071919:lean_proved, oeis_catalog:A074246:lean_proved, oeis_catalog:A075497:lean_proved, oeis_catalog:A075498:lean_proved, oeis_catalog:A079508:lean_proved, oeis_catalog:A079510:lean_proved, oeis_catalog:A079621:lean_proved, oeis_catalog:A079638:lean_proved, oeis_catalog:A080248:lean_proved, oeis_catalog:A081577:lean_proved, oeis_catalog:A081578:lean_proved, oeis_catalog:A081579:lean_proved, oeis_catalog:A081580:lean_proved, oeis_catalog:A081581:lean_proved, oeis_catalog:A081582:lean_proved, oeis_catalog:A082649:lean_proved, oeis_catalog:A082985:lean_proved, oeis_catalog:A084534:lean_proved, oeis_catalog:A084950:lean_proved, oeis_catalog:A085841:lean_proved, oeis_catalog:A085881:lean_proved, oeis_catalog:A087727:lean_proved, oeis_catalog:A088996:lean_proved, oeis_catalog:A089231:lean_proved, oeis_catalog:A090452:lean_proved, oeis_catalog:A090582:lean_proved, oeis_catalog:A091044:lean_proved, oeis_catalog:A092879:lean_proved, oeis_catalog:A093127:lean_proved, oeis_catalog:A093375:lean_proved, oeis_catalog:A093560:lean_proved, oeis_catalog:A093561:lean_proved, oeis_catalog:A093562:lean_proved, oeis_catalog:A093563:lean_proved, oeis_catalog:A093564:lean_proved, oeis_catalog:A093565:lean_proved, oeis_catalog:A093644:lean_proved, oeis_catalog:A093645:lean_proved, oeis_catalog:A094262:lean_proved, oeis_catalog:A094305:lean_proved, oeis_catalog:A094503:lean_proved, oeis_catalog:A094638:lean_proved, oeis_catalog:A094645:lean_proved, oeis_catalog:A094646:lean_proved, oeis_catalog:A097805:lean_proved, oeis_catalog:A097806:lean_proved, oeis_catalog:A099040:lean_proved, oeis_catalog:A099093:lean_proved, oeis_catalog:A099605:lean_proved, oeis_catalog:A100861:lean_proved, oeis_catalog:A100862:lean_proved, oeis_catalog:A100898:lean_proved, oeis_catalog:A101280:lean_proved, oeis_catalog:A101343:lean_proved, oeis_catalog:A101350:lean_proved, oeis_catalog:A102365:lean_proved, oeis_catalog:A102413:lean_proved, oeis_catalog:A102547:lean_proved, oeis_catalog:A102662:lean_proved, oeis_catalog:A103327:lean_proved, oeis_catalog:A103406:lean_proved, oeis_catalog:A103450:lean_proved, oeis_catalog:A103633:lean_proved, oeis_catalog:A104029:lean_proved, oeis_catalog:A104978:lean_proved, oeis_catalog:A105070:lean_proved, oeis_catalog:A105793:lean_proved, oeis_catalog:A105868:lean_proved, oeis_catalog:A106246:lean_proved, oeis_catalog:A106828:lean_proved, oeis_catalog:A107131:lean_proved, oeis_catalog:A108426:lean_proved, oeis_catalog:A109446:lean_proved, oeis_catalog:A109447:lean_proved, oeis_catalog:A109692:lean_proved, oeis_catalog:A109955:lean_proved, oeis_catalog:A109960:lean_proved, oeis_catalog:A109983:lean_proved, oeis_catalog:A110200:lean_proved, oeis_catalog:A110205:lean_proved, oeis_catalog:A110441:lean_proved, oeis_catalog:A110552:lean_proved, oeis_catalog:A111006:lean_proved, oeis_catalog:A111049:lean_proved, oeis_catalog:A111910:lean_proved, oeis_catalog:A112007:lean_proved, oeis_catalog:A112486:lean_proved, oeis_catalog:A113214:lean_proved, oeis_catalog:A113413:lean_proved, oeis_catalog:A113953:lean_proved, oeis_catalog:A114123:lean_proved, oeis_catalog:A114580:lean_proved, oeis_catalog:A114687:lean_proved, oeis_catalog:A114690:lean_proved, oeis_catalog:A115951:lean_proved, oeis_catalog:A116088:lean_proved, oeis_catalog:A116089:lean_proved, oeis_catalog:A116412:lean_proved, oeis_catalog:A116414:lean_proved, oeis_catalog:A116666:lean_proved, oeis_catalog:A117317:lean_proved, oeis_catalog:A118394:lean_proved, oeis_catalog:A118424:lean_proved, oeis_catalog:A118884:lean_proved, oeis_catalog:A118931:lean_proved, oeis_catalog:A118933:lean_proved, oeis_catalog:A119308:lean_proved, oeis_catalog:A119462:lean_proved, oeis_catalog:A119808:lean_proved, oeis_catalog:A119900:lean_proved, oeis_catalog:A120095:lean_proved, oeis_catalog:A120987:lean_proved, oeis_catalog:A121314:lean_proved, oeis_catalog:A121574:lean_proved, oeis_catalog:A122848:lean_proved, oeis_catalog:A123097:lean_proved, oeis_catalog:A123125:lean_proved, oeis_catalog:A124038:lean_proved, oeis_catalog:A124324:lean_proved, oeis_catalog:A124376:lean_proved, oeis_catalog:A124848:lean_proved, oeis_catalog:A124860:lean_proved, oeis_catalog:A124932:lean_proved, oeis_catalog:A125092:lean_proved, oeis_catalog:A125662:lean_proved, oeis_catalog:A126216:lean_proved, oeis_catalog:A126351:lean_proved, oeis_catalog:A127647:lean_proved, oeis_catalog:A127733:lean_proved, oeis_catalog:A127773:lean_proved, oeis_catalog:A127893:lean_proved, oeis_catalog:A127952:lean_proved, oeis_catalog:A128099:lean_proved, oeis_catalog:A128100:lean_proved, oeis_catalog:A128540:lean_proved, oeis_catalog:A128541:lean_proved, oeis_catalog:A128813:lean_proved, oeis_catalog:A128908:lean_proved, oeis_catalog:A129186:lean_proved, oeis_catalog:A129710:lean_proved, oeis_catalog:A129718:lean_proved, oeis_catalog:A130123:lean_proved, oeis_catalog:A130534:lean_proved, oeis_catalog:A130850:lean_proved, oeis_catalog:A131222:lean_proved, oeis_catalog:A131431:lean_proved, oeis_catalog:A131436:lean_proved, oeis_catalog:A132200:lean_proved, oeis_catalog:A132393:lean_proved, oeis_catalog:A132774:lean_proved, oeis_catalog:A132812:lean_proved, oeis_catalog:A132885:lean_proved, oeis_catalog:A132918:lean_proved, oeis_catalog:A133080:lean_proved, oeis_catalog:A133081:lean_proved, oeis_catalog:A133087:lean_proved, oeis_catalog:A133109:lean_proved, oeis_catalog:A134058:lean_proved, oeis_catalog:A134059:lean_proved, oeis_catalog:A134082:lean_proved, oeis_catalog:A134239:lean_proved, oeis_catalog:A134309:lean_proved, oeis_catalog:A134346:lean_proved, oeis_catalog:A134434:lean_proved, oeis_catalog:A135065:lean_proved, oeis_catalog:A135089:lean_proved, oeis_catalog:A135838:lean_proved, oeis_catalog:A136011:lean_proved, oeis_catalog:A136124:lean_proved, oeis_catalog:A136158:lean_proved, oeis_catalog:A136394:lean_proved, oeis_catalog:A136521:lean_proved, oeis_catalog:A136572:lean_proved, oeis_catalog:A137320:lean_proved, oeis_catalog:A137338:lean_proved, oeis_catalog:A137597:lean_proved, oeis_catalog:A139548:lean_proved, oeis_catalog:A141611:lean_proved, oeis_catalog:A141697:lean_proved, oeis_catalog:A142071:lean_proved, oeis_catalog:A142458:lean_proved, oeis_catalog:A142460:lean_proved, oeis_catalog:A142461:lean_proved, oeis_catalog:A142462:lean_proved, oeis_catalog:A142963:lean_proved, oeis_catalog:A143007:lean_proved, oeis_catalog:A143683:lean_proved, oeis_catalog:A143685:lean_proved, oeis_catalog:A144084:lean_proved, oeis_catalog:A144090:lean_proved, oeis_catalog:A144299:lean_proved, oeis_catalog:A144436:lean_proved, oeis_catalog:A144438:lean_proved, oeis_catalog:A144439:lean_proved, oeis_catalog:A144440:lean_proved, oeis_catalog:A144441:lean_proved, oeis_catalog:A144442:lean_proved, oeis_catalog:A144443:lean_proved, oeis_catalog:A144445:lean_proved, oeis_catalog:A144696:lean_proved, oeis_catalog:A144697:lean_proved, oeis_catalog:A144698:lean_proved, oeis_catalog:A144699:lean_proved, oeis_catalog:A145324:lean_proved, oeis_catalog:A145596:lean_proved, oeis_catalog:A145597:lean_proved, oeis_catalog:A146986:lean_proved, oeis_catalog:A146987:lean_proved, oeis_catalog:A146988:lean_proved, oeis_catalog:A147565:lean_proved, oeis_catalog:A147644:lean_proved, oeis_catalog:A147716:lean_proved, oeis_catalog:A151775:lean_proved, oeis_catalog:A152198:lean_proved, oeis_catalog:A152815:lean_proved, oeis_catalog:A152828:lean_proved, oeis_catalog:A152830:lean_proved, oeis_catalog:A152842:lean_proved, oeis_catalog:A154227:lean_proved, oeis_catalog:A154228:lean_proved, oeis_catalog:A154233:lean_proved, oeis_catalog:A154929:lean_proved, oeis_catalog:A154986:lean_proved, oeis_catalog:A155112:lean_proved, oeis_catalog:A156308:lean_proved, oeis_catalog:A156319:lean_proved, oeis_catalog:A156654:lean_proved, oeis_catalog:A156992:lean_proved, oeis_catalog:A157012:lean_proved, oeis_catalog:A157147:lean_proved, oeis_catalog:A157148:lean_proved, oeis_catalog:A157149:lean_proved, oeis_catalog:A157150:lean_proved, oeis_catalog:A157151:lean_proved, oeis_catalog:A157169:lean_proved, oeis_catalog:A157171:lean_proved, oeis_catalog:A157637:lean_proved, oeis_catalog:A160562:lean_proved, oeis_catalog:A163932:lean_proved, oeis_catalog:A163936:lean_proved, oeis_catalog:A163937:lean_proved, oeis_catalog:A165253:lean_proved, oeis_catalog:A165891:lean_proved, oeis_catalog:A166345:lean_proved, oeis_catalog:A166960:lean_proved, oeis_catalog:A166961:lean_proved, oeis_catalog:A166962:lean_proved, oeis_catalog:A166972:lean_proved, oeis_catalog:A167194:lean_proved, oeis_catalog:A167884:lean_proved, oeis_catalog:A169803:lean_proved, oeis_catalog:A171608:lean_proved, oeis_catalog:A172185:lean_proved, oeis_catalog:A172431:lean_proved, oeis_catalog:A176230:lean_proved, oeis_catalog:A176331:lean_proved, oeis_catalog:A177040:lean_proved, oeis_catalog:A178820:lean_proved, oeis_catalog:A178821:lean_proved, oeis_catalog:A178822:lean_proved, oeis_catalog:A179900:lean_proved, oeis_catalog:A180013:lean_proved, oeis_catalog:A181289:lean_proved, oeis_catalog:A181327:lean_proved, oeis_catalog:A181371:lean_proved, oeis_catalog:A181543:lean_proved, oeis_catalog:A184883:lean_proved, oeis_catalog:A185263:lean_proved, oeis_catalog:A185410:lean_proved, oeis_catalog:A185411:lean_proved, oeis_catalog:A186370:lean_proved, oeis_catalog:A188440:lean_proved, oeis_catalog:A190088:lean_proved, oeis_catalog:A190152:lean_proved, oeis_catalog:A191935:lean_proved, oeis_catalog:A193636:lean_proved, oeis_catalog:A193685:lean_proved, oeis_catalog:A193722:lean_proved, oeis_catalog:A193726:lean_proved, oeis_catalog:A193727:lean_proved, oeis_catalog:A193728:lean_proved, oeis_catalog:A193729:lean_proved, oeis_catalog:A193730:lean_proved, oeis_catalog:A193731:lean_proved, oeis_catalog:A193734:lean_proved, oeis_catalog:A193735:lean_proved, oeis_catalog:A196347:lean_proved, oeis_catalog:A198954:lean_proved, oeis_catalog:A200091:lean_proved, oeis_catalog:A200536:lean_proved, oeis_catalog:A201013:lean_proved, oeis_catalog:A201701:lean_proved, oeis_catalog:A201972:lean_proved, oeis_catalog:A202023:lean_proved, oeis_catalog:A202064:lean_proved, oeis_catalog:A202241:lean_proved, oeis_catalog:A204420:lean_proved, oeis_catalog:A206294:lean_proved, oeis_catalog:A206306:lean_proved, oeis_catalog:A207607:lean_proved, oeis_catalog:A207823:lean_proved, oeis_catalog:A207824:lean_proved, oeis_catalog:A208509:lean_proved, oeis_catalog:A208513:lean_proved, oeis_catalog:A209414:lean_proved, oeis_catalog:A209599:lean_proved, oeis_catalog:A210039:lean_proved, oeis_catalog:A211956:lean_proved, oeis_catalog:A216182:lean_proved, oeis_catalog:A219512:lean_proved, oeis_catalog:A223168:lean_proved, oeis_catalog:A225470:lean_proved, oeis_catalog:A225471:lean_proved, oeis_catalog:A225476:lean_proved, oeis_catalog:A228955:lean_proved, oeis_catalog:A229032:lean_proved, oeis_catalog:A232535:lean_proved, oeis_catalog:A236076:lean_proved, oeis_catalog:A236471:lean_proved, oeis_catalog:A241171:lean_proved, oeis_catalog:A245962:lean_proved, oeis_catalog:A245963:lean_proved, oeis_catalog:A245964:lean_proved, oeis_catalog:A249139:lean_proved, oeis_catalog:A249247:lean_proved, oeis_catalog:A249248:lean_proved, oeis_catalog:A249250:lean_proved, oeis_catalog:A249251:lean_proved, oeis_catalog:A249252:lean_proved, oeis_catalog:A249253:lean_proved, oeis_catalog:A253284:lean_proved, oeis_catalog:A257606:lean_proved, oeis_catalog:A257607:lean_proved, oeis_catalog:A257608:lean_proved, oeis_catalog:A257609:lean_proved, oeis_catalog:A257610:lean_proved, oeis_catalog:A257611:lean_proved, oeis_catalog:A257612:lean_proved, oeis_catalog:A257613:lean_proved, oeis_catalog:A257614:lean_proved, oeis_catalog:A257615:lean_proved, oeis_catalog:A257616:lean_proved, oeis_catalog:A257617:lean_proved, oeis_catalog:A257618:lean_proved, oeis_catalog:A257619:lean_proved, oeis_catalog:A257620:lean_proved, oeis_catalog:A257621:lean_proved, oeis_catalog:A257622:lean_proved, oeis_catalog:A257623:lean_proved, oeis_catalog:A257624:lean_proved, oeis_catalog:A257625:lean_proved, oeis_catalog:A257626:lean_proved, oeis_catalog:A257627:lean_proved, oeis_catalog:A258758:lean_proved, oeis_catalog:A261139:lean_proved, oeis_catalog:A261318:lean_proved, oeis_catalog:A261711:lean_proved, oeis_catalog:A262226:lean_proved, oeis_catalog:A262704:lean_proved, oeis_catalog:A263789:lean_proved, oeis_catalog:A265644:lean_proved, oeis_catalog:A266213:lean_proved, oeis_catalog:A266326:lean_proved, oeis_catalog:A267006:lean_proved, oeis_catalog:A267482:lean_proved, oeis_catalog:A268434:lean_proved, oeis_catalog:A269944:lean_proved, oeis_catalog:A269947:lean_proved, oeis_catalog:A272866:lean_proved, oeis_catalog:A272867:lean_proved, oeis_catalog:A273717:lean_proved, oeis_catalog:A274496:lean_proved, oeis_catalog:A274498:lean_proved, oeis_catalog:A275442:lean_proved, oeis_catalog:A281260:lean_proved, oeis_catalog:A282629:lean_proved, oeis_catalog:A285061:lean_proved, oeis_catalog:A286718:lean_proved, oeis_catalog:A288874:lean_proved, oeis_catalog:A290319:lean_proved, oeis_catalog:A297191:lean_proved, oeis_catalog:A298668:lean_proved, oeis_catalog:A304236:lean_proved, oeis_catalog:A304330:lean_proved, oeis_catalog:A304334:lean_proved, oeis_catalog:A304429:lean_proved, oeis_catalog:A305540:lean_proved, oeis_catalog:A306364:lean_proved, oeis_catalog:A307419:lean_proved, oeis_catalog:A308737:lean_proved, oeis_catalog:A316649:lean_proved, oeis_catalog:A317496:lean_proved, oeis_catalog:A317497:lean_proved, oeis_catalog:A318107:lean_proved, oeis_catalog:A318397:lean_proved, oeis_catalog:A318772:lean_proved, oeis_catalog:A318773:lean_proved, oeis_catalog:A321434:lean_proved, oeis_catalog:A321966:lean_proved, oeis_catalog:A327997:lean_proved, oeis_catalog:A330381:lean_proved, oeis_catalog:A330509:lean_proved, oeis_catalog:A330609:lean_proved, oeis_catalog:A330798:lean_proved, oeis_catalog:A331333:lean_proved, oeis_catalog:A334009:lean_proved, oeis_catalog:A339000:lean_proved, oeis_catalog:A340556:lean_proved, oeis_catalog:A341287:lean_proved, oeis_catalog:A345013:lean_proved, oeis_catalog:A347056:lean_proved, oeis_catalog:A347823:lean_proved, oeis_catalog:A350512:lean_proved, oeis_catalog:A355144:lean_proved, oeis_catalog:A356546:lean_proved, oeis_catalog:A357367:lean_proved, oeis_catalog:A357613:lean_proved, oeis_catalog:A358628:lean_proved, oeis_catalog:A358735:lean_proved, oeis_catalog:A359363:lean_proved, oeis_catalog:A360282:lean_proved, oeis_catalog:A361357:lean_proved, oeis_catalog:A366159:lean_proved, oeis_catalog:A367177:lean_proved, oeis_catalog:A367270:lean_proved, oeis_catalog:A367873:lean_proved, oeis_catalog:A371080:lean_proved, oeis_catalog:A371081:lean_proved, oeis_catalog:A371259:lean_proved, oeis_catalog:A371898:lean_proved, oeis_catalog:A372868:lean_proved, oeis_catalog:A372873:lean_proved, oeis_catalog:A376827:lean_proved, oeis_catalog:A380114:lean_proved, oeis_catalog:A380860:lean_proved, oeis_catalog:A386789:lean_proved, oeis_catalog:A390883:lean_proved, oeis_catalog:A391403:lean_proved, oeis_catalog:A394446:lean_proved, oeis_catalog:A394813:lean_proved, oeis_catalog:A395326:lean_proved, oeis_catalog:A397048:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2 for every n >= 0 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}