{"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":[["0","1","1"],["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["520","-352","-1880","-184","824"],["1246","906","-1832","68","1604"],["923","2096","1304","912","950"],["197","966","1707","1064","149"],["0","128","495","500","-28"],["0","0","44","104","-7"],["0","0","0","8","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-520","-2104","-8416","-14392","-8384","-824"],["-744","-1766","-12018","-24848","-13940","44"],["-224","721","-3240","-11128","-6016","1434"],["0","383","522","-183","168","559"],["0","0","160","557","852","-28"],["0","0","0","68","248","-21"],["0","0","0","0","24","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["18","-866","2114","14550","19616","8476","412"],["27","-1722","-2403","774","-786","-4958","-2700"],["0","-532","-1034","-4432","-9612","-6734","-883"],["0","0","45","-1068","-3438","-2122","226"],["0","0","0","-80","-521","-238","42"],["0","0","0","0","-36","24","-7"],["0","0","0","0","0","8","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","14","-298","3886","16906","23040","12392","2060"],["0","21","-802","1137","10540","14790","6306","96"],["0","0","-252","-43","1804","2372","374","-403"],["0","0","0","32","134","-171","-294","-21"],["0","0","0","0","0","-120","-106","14"],["0","0","0","0","0","-16","-16","0"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","2","-24","686","2678","3604","2050","412"],["0","0","3","-88","359","2032","2804","1406","184"],["0","0","0","-28","67","598","876","366","-7"],["0","0","0","0","12","104","165","66","-7"],["0","0","0","0","0","8","16","8","0"]]},"deriv_order":4,"shift":0},{"coeff":{"coeffs":[["484","484"],["654","960"],["170","584"],["0","108"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","1452","1452"],["0","1962","2880"],["0","510","1752"],["0","0","324"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","484","484"],["0","0","654","960"],["0","0","170","584"],["0","0","0","108"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0","4","9"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["314","116"],["170","260"],["0","108"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0","0"],["9","900","1079","92","-228"],["-332","34","1287","200","-405"],["-197","-454","-167","56","-191"],["0","-128","-275","-100","-7"],["0","0","-44","-56","7"],["0","0","0","-8","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["74","508","478","448","860","456"],["56","356","936","1014","732","354"],["0","52","370","558","268","28"],["0","0","56","152","82","-14"],["0","0","0","16","16","0"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-36","-382","-1148","-1527","-952","-228"],["-3","32","-261","-1116","-1483","-840","-177"],["0","28","-31","-334","-477","-216","-14"],["0","0","-12","-72","-101","-34","7"],["0","0","0","-8","-16","-8","0"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0","1","1"],["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":0,"offset":2,"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":[["1","-1","-2"],["1","0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-1","1","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["484","3388","10164","16940","16940","10164","3388","484"],["654","4884","15570","27480","29010","18324","6414","960"],["170","1604","6054","12160","14230","9780","3674","584"],["0","108","648","1620","2160","1620","648","108"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1452","10164","30492","50820","50820","30492","10164","1452"],["0","1962","14652","46710","82440","87030","54972","19242","2880"],["0","510","4812","18162","36480","42690","29340","11022","1752"],["0","0","324","1944","4860","6480","4860","1944","324"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","484","3388","10164","16940","16940","10164","3388","484"],["0","0","654","4884","15570","27480","29010","18324","6414","960"],["0","0","170","1604","6054","12160","14230","9780","3674","584"],["0","0","0","108","648","1620","2160","1620","648","108"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["1","2","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["2","3","-3","-7","-3"],["1","2","0","-2","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1","-3","-2","2","3","1"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-1","-2","1","4","1","-2","-1"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["2"],["3"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-2"],["-2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["18","658","7954","39178","103026","162470","160538","99054","36148","6736","412"],["27","376","4181","23158","68985","121156","130571","85686","31988","5560","184"],["0","20","483","3694","13134","25758","29780","20298","7554","1174","-7"],["0","0","44","344","1133","2038","2155","1324","419","38","-7"],["0","0","0","8","48","120","160","120","48","8","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","14","238","5718","36254","110246","193890","211438","144778","60084","13576","1236"],["0","21","-66","2407","22078","76131","141548","157289","106422","41968","8386","552"],["0","0","-76","281","4812","18694","36424","40884","26876","9602","1404","-21"],["0","0","0","84","744","2679","5154","5745","3684","1209","114","-21"],["0","0","0","0","24","144","360","480","360","144","24","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","2","-16","602","5286","18338","35254","41634","31042","14276","3698","412"],["0","0","3","-76","25","2952","12737","26162","31119","22420","9532","2142","184"],["0","0","0","-28","-45","698","3558","7698","9172","6270","2298","338","-7"],["0","0","0","0","12","152","653","1398","1675","1132","387","38","-7"],["0","0","0","0","0","8","48","120","160","120","48","8","0"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["1","4","6","4","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["6","24","36","24","6"],["5","20","30","20","5"],["1","4","6","4","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-4","-20","-40","-40","-20","-4"],["-2","-10","-20","-20","-10","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["1","6","15","20","15","6","1"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["-3","-15","-30","-30","-15","-3","0"],["-1","-4","-5","0","5","4","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["1","5","9","5","-5","-9","-5","-1"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1","6","15","20","15","6","1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":0,"shift":0},{"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":{"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":[["0","1","1"],["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2","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":"periodic_product","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":"c and the seeds are nonzero, real-rooted and of one sign, so P_(n0 + k m + i) = c^m P_(n0 + i) is too","route":"periodic_product","success":true,"variant":"P_(n+k) = c(x) P_n"}],"conditional_on":[],"route":"periodic_product","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"}