{"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":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["0","-1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","1"],["0","2"],["0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","3"],["0","0","6"],["0","0","3"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","1"],["0","0","0","2"],["0","0","0","1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0"],["0","3"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","0"],["0","0","1"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["0","0"],["0","-1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["0","-1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0"],["0","0","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","0","3"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","1"]]},"deriv_order":2}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["1"],["-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","0","1"],["0","0","0","3"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","1"],["0","0","0","0","2"]]},"deriv_order":1}],"row_multiplier":["0","0","1"]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["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 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 1 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"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 [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + 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":"1","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":1,"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":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["0","-1"]]},"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 >= 1"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 1, 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 >= 1, 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 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":"no theorem matched (weighted_renewal_window: coefficients must be index-independent; 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: 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 >= 2, with a_n(x) = (0) + (-n + 1) x and b_n(x) = (0) + (0) x + (0) x^2; open hypotheses: degree_step_positive: fails at n = 2: lambda_n = -n + 1 = -1 there (exact); constant_term_positive: fails at n = 2: a_n(0) = 0 = 0 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":"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":"no lean_proved catalog entry equals the parent; 68 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A003506:lean_proved, oeis_catalog:A008277:lean_proved, oeis_catalog:A008278:lean_proved, oeis_catalog:A008288:lean_proved, oeis_catalog:A008292:lean_proved, oeis_catalog:A019538:lean_proved, oeis_catalog:A026729:lean_proved, oeis_catalog:A035607:lean_proved, oeis_catalog:A039755:lean_proved, oeis_catalog:A046716:lean_proved, oeis_catalog:A048854:lean_proved, oeis_catalog:A052553:lean_proved, oeis_catalog:A056242:lean_proved, oeis_catalog:A059110:lean_proved, oeis_catalog:A060187:lean_proved, oeis_catalog:A062145:lean_proved, oeis_catalog:A063007:lean_proved, oeis_catalog:A063967:lean_proved, oeis_catalog:A071951:lean_proved, oeis_catalog:A078812:lean_proved, oeis_catalog:A085478:lean_proved, oeis_catalog:A086645:lean_proved, oeis_catalog:A089627:lean_proved, oeis_catalog:A091042:lean_proved, oeis_catalog:A098158:lean_proved, oeis_catalog:A098172:lean_proved, oeis_catalog:A099097:lean_proved, oeis_catalog:A104684:lean_proved, oeis_catalog:A105278:lean_proved, oeis_catalog:A106800:lean_proved, oeis_catalog:A111125:lean_proved, oeis_catalog:A114656:lean_proved, oeis_catalog:A122542:lean_proved, oeis_catalog:A123160:lean_proved, oeis_catalog:A127648:lean_proved, oeis_catalog:A131689:lean_proved, oeis_catalog:A134991:lean_proved, oeis_catalog:A142075:lean_proved, oeis_catalog:A142459:lean_proved, oeis_catalog:A143395:lean_proved, oeis_catalog:A154602:lean_proved, oeis_catalog:A156289:lean_proved, oeis_catalog:A158909:lean_proved, oeis_catalog:A166973:lean_proved, oeis_catalog:A173018:lean_proved, oeis_catalog:A201637:lean_proved, oeis_catalog:A201730:lean_proved, oeis_catalog:A219836:lean_proved, oeis_catalog:A225117:lean_proved, oeis_catalog:A225466:lean_proved, oeis_catalog:A225467:lean_proved, oeis_catalog:A225468:lean_proved, oeis_catalog:A225469:lean_proved, oeis_catalog:A225472:lean_proved, oeis_catalog:A225473:lean_proved, oeis_catalog:A235706:lean_proved, oeis_catalog:A253283:lean_proved, oeis_catalog:A259454:lean_proved, oeis_catalog:A269939:lean_proved, oeis_catalog:A269945:lean_proved, oeis_catalog:A271697:lean_proved, oeis_catalog:A271703:lean_proved, oeis_catalog:A271705:lean_proved, oeis_catalog:A286724:lean_proved, oeis_catalog:A292219:lean_proved, oeis_catalog:A321331:lean_proved, oeis_catalog:A334218:lean_proved, oeis_catalog:A390433: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 >= 1, 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 >= 1 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}