{"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"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A007318 of Riordan array (1, 2+2x)."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-72","-696","-2144","-2768","-1312","152","216"],["0","-216","-1212","-2252","-1548","-112","180"],["0","0","-156","-544","-570","-134","48"],["0","0","0","-36","-72","-32","4"],["0","0","0","0","-2","-2","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-216","-2088","-6432","-8304","-3936","456","648"],["0","0","-648","-3636","-6756","-4644","-336","540"],["0","0","0","-468","-1632","-1710","-402","144"],["0","0","0","0","-108","-216","-96","12"],["0","0","0","0","0","-6","-6","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-72","-696","-2144","-2768","-1312","152","216"],["0","0","0","-216","-1212","-2252","-1548","-112","180"],["0","0","0","0","-156","-544","-570","-134","48"],["0","0","0","0","0","-36","-72","-32","4"],["0","0","0","0","0","0","-2","-2","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-72","-696","-1892","-2036","-640","308","180"],["0","-216","-1128","-1900","-1108","36","156"],["0","0","-156","-508","-486","-90","44"],["0","0","0","-36","-68","-28","4"],["0","0","0","0","-2","-2","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-216","-2088","-5676","-6108","-1920","924","540"],["0","0","-648","-3384","-5700","-3324","108","468"],["0","0","0","-468","-1524","-1458","-270","132"],["0","0","0","0","-108","-204","-84","12"],["0","0","0","0","0","-6","-6","0"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","-72","-696","-1892","-2036","-640","308","180"],["0","0","0","-216","-1128","-1900","-1108","36","156"],["0","0","0","0","-156","-508","-486","-90","44"],["0","0","0","0","0","-36","-68","-28","4"],["0","0","0","0","0","0","-2","-2","0"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["36","204","340","172","-40","-40"],["0","108","342","294","14","-48"],["0","0","78","140","43","-18"],["0","0","0","18","14","-2"],["0","0","0","0","1","0"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","108","612","1020","516","-120","-120"],["0","0","324","1026","882","42","-144"],["0","0","0","234","420","129","-54"],["0","0","0","0","54","42","-6"],["0","0","0","0","0","3","0"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["0","0","36","204","340","172","-40","-40"],["0","0","0","108","342","294","14","-48"],["0","0","0","0","78","140","43","-18"],["0","0","0","0","0","18","14","-2"],["0","0","0","0","0","0","1","0"]]},"deriv_order":2,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["2","8"]],"recurrence":{"denominator":{"coeffs":[["36","-12","-32","0","8","0"],["0","108","30","-50","-22","0"],["0","0","78","32","-17","-6"],["0","0","0","18","6","-2"],["0","0","0","0","1","0"]]},"terms":[{"coeff":{"coeffs":[["72","264","344","152","-48","-64","-16"],["0","216","588","508","68","-104","-36"],["0","0","156","328","186","-10","-24"],["0","0","0","36","56","16","-4"],["0","0","0","0","2","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["72","264","260","-20","-144","-68","-12"],["0","216","504","300","-84","-124","-28"],["0","0","156","292","126","-30","-20"],["0","0","0","36","52","12","-4"],["0","0","0","0","2","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1","3","3","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"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["1296","13392","57888","138720","202528","182816","94272","16448","-10160","-7152","-1760","-160"],["3888","44064","204120","518328","798384","762792","426224","98408","-29760","-27208","-7144","-672"],["3888","51840","267840","740808","1229700","1266008","778332","228880","-21172","-39240","-11420","-1128"],["1296","25056","157680","500304","927604","1057400","729932","267104","15740","-25672","-9020","-960"],["0","3888","38880","158544","350568","462000","368280","165696","29208","-6336","-3576","-432"],["0","0","2808","20088","60408","98936","94920","52552","14504","552","-608","-96"],["0","0","0","648","3636","8472","10508","7344","2764","440","-12","-8"],["0","0","0","0","36","168","316","304","156","40","4","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","3888","40176","173664","416160","607584","548448","282816","49344","-30480","-21456","-5280","-480"],["0","11664","132192","612360","1554984","2395152","2288376","1278672","295224","-89280","-81624","-21432","-2016"],["0","11664","155520","803520","2222424","3689100","3798024","2334996","686640","-63516","-117720","-34260","-3384"],["0","3888","75168","473040","1500912","2782812","3172200","2189796","801312","47220","-77016","-27060","-2880"],["0","0","11664","116640","475632","1051704","1386000","1104840","497088","87624","-19008","-10728","-1296"],["0","0","0","8424","60264","181224","296808","284760","157656","43512","1656","-1824","-288"],["0","0","0","0","1944","10908","25416","31524","22032","8292","1320","-36","-24"],["0","0","0","0","0","108","504","948","912","468","120","12","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","1296","13392","57888","138720","202528","182816","94272","16448","-10160","-7152","-1760","-160"],["0","0","3888","44064","204120","518328","798384","762792","426224","98408","-29760","-27208","-7144","-672"],["0","0","3888","51840","267840","740808","1229700","1266008","778332","228880","-21172","-39240","-11420","-1128"],["0","0","1296","25056","157680","500304","927604","1057400","729932","267104","15740","-25672","-9020","-960"],["0","0","0","3888","38880","158544","350568","462000","368280","165696","29208","-6336","-3576","-432"],["0","0","0","0","2808","20088","60408","98936","94920","52552","14504","552","-608","-96"],["0","0","0","0","0","648","3636","8472","10508","7344","2764","440","-12","-8"],["0","0","0","0","0","0","36","168","316","304","156","40","4","0"]]},"deriv_order":2}],"shift":0}],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-2","-2"],["2","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-6","-8","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","15120","107568","329040","559424","564832","324000","72928","-26560","-22896","-5968","-560"],["0","0","45360","336960","1075824","1912080","2030560","1251760","346688","-57744","-74192","-20848","-2032"],["0","0","45360","365472","1256976","2399832","2747552","1864264","637424","-1816","-83696","-26920","-2784"],["0","0","15120","150336","606384","1321472","1708624","1324128","563248","76352","-34656","-15328","-1760"],["0","0","0","14256","99936","294864","473680","444400","238160","60912","-272","-3424","-496"],["0","0","0","0","3744","20784","47888","58672","40400","14864","2224","-112","-48"],["0","0","0","0","0","216","1008","1896","1824","936","240","24","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-18144","-142128","-487728","-954960","-1157120","-864928","-347808","-11488","62560","31440","6736","560"],["0","0","-36288","-293328","-1036368","-2087040","-2605744","-2027984","-892048","-107376","101168","58208","12832","1072"],["0","0","-18144","-160272","-606096","-1285872","-1674320","-1361792","-656528","-144608","16560","17488","3744","272"],["0","0","0","-9072","-54000","-128304","-148224","-73280","2656","7168","-13312","-13136","-4208","-464"],["0","0","0","0","3456","27648","89280","152192","147008","77312","16192","-2944","-1984","-256"],["0","0","0","0","0","2160","11952","27408","33328","22640","8080","1072","-112","-32"],["0","0","0","0","0","0","144","672","1264","1216","624","160","16","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","-7776","-65232","-239760","-503280","-655744","-532064","-241632","-25760","34400","20016","4592","400"],["0","0","0","-15552","-138672","-540432","-1201824","-1663056","-1449296","-739952","-145072","59856","46144","11456","1040"],["0","0","0","-7776","-81648","-364176","-910224","-1403232","-1368640","-814016","-244096","7616","31408","9296","912"],["0","0","0","0","-8208","-66096","-228240","-441088","-519584","-377184","-157856","-27520","4272","2576","304"],["0","0","0","0","0","-2592","-16704","-45984","-70112","-63968","-34912","-10464","-1184","128","32"],["0","0","0","0","0","0","-144","-816","-1936","-2480","-1840","-784","-176","-16","0"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["6","14","10","2"],["12","28","20","4"],["6","14","10","2"]]},"step_quotients":[[{"coeff":{"coeffs":[["24","68","68","28","4"],["48","136","136","56","8"],["24","68","68","28","4"]]},"deriv_order":0},{"coeff":{"coeffs":[["-36","-132","-184","-120","-36","-4"],["-36","-132","-184","-120","-36","-4"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-36","-96","-88","-32","-4"],["-36","-96","-88","-32","-4"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-1","-2"],["1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["6","10","4"],["-6","-8","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["6","14","10","2"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","-18144","-122688","-357408","-579648","-557808","-301088","-55824","32768","24016","6048","560"],["0","0","-36288","-260928","-809136","-1401960","-1456480","-879480","-234864","46120","54144","15096","1472"],["0","0","-18144","-153792","-549936","-1085712","-1285328","-909264","-337200","-17968","34800","12240","1312"],["0","0","0","-15552","-102096","-284352","-433456","-386336","-194928","-44160","2992","3360","448"],["0","0","0","0","-3888","-21168","-47808","-57360","-38592","-13776","-1920","144","48"],["0","0","0","0","0","-216","-1008","-1896","-1824","-936","-240","-24","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-7776","-57456","-182304","-320976","-334768","-197296","-44336","18576","15824","4192","400"],["0","0","0","-15552","-123120","-417312","-784512","-878544","-570752","-169200","24128","35728","10416","1040"],["0","0","0","-7776","-73872","-290304","-619920","-783312","-585328","-228688","-15408","23024","8384","912"],["0","0","0","0","-8208","-57888","-170352","-270736","-248848","-128336","-29520","2000","2272","304"],["0","0","0","0","0","-2592","-14112","-31872","-38240","-25728","-9184","-1280","96","32"],["0","0","0","0","0","0","-144","-672","-1264","-1216","-624","-160","-16","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["36","168","316","304","156","40","4"],["108","504","948","912","468","120","12"],["108","504","948","912","468","120","12"],["36","168","316","304","156","40","4"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","-72","-336","-632","-608","-312","-80","-8"],["36","-48","-692","-1592","-1668","-896","-236","-24"],["108","288","-60","-984","-1356","-816","-228","-24"],["108","432","612","280","-140","-192","-68","-8"],["36","168","316","304","156","40","4","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","72","408","968","1240","920","392","88","8"],["-216","-1152","-2496","-2752","-1520","-256","128","64","8"],["-432","-2520","-6216","-8408","-6760","-3272","-920","-136","-8"],["-216","-1296","-3312","-4688","-4000","-2096","-656","-112","-8"]]},"deriv_order":1},{"coeff":{"coeffs":[["216","1512","4608","8000","8688","6096","2752","768","120","8"],["432","3024","9216","16000","17376","12192","5504","1536","240","16"],["216","1512","4608","8000","8688","6096","2752","768","120","8"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["-72","-264","-296","24","296","232","72","8"],["-216","-864","-1224","-560","280","384","136","16"],["-216","-936","-1560","-1192","-328","72","56","8"],["-72","-336","-632","-608","-312","-80","-8","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["216","1296","3312","4688","4000","2096","656","112","8"],["432","2592","6624","9376","8000","4192","1312","224","16"],["216","1296","3312","4688","4000","2096","656","112","8"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["36","168","316","304","156","40","4"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-2","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-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 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"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","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":"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 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":"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 >= 2, with A_n(x) = (2) + (2) x, V_n(x) = 0, W_n(x) = (2) + (2) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = 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 = 2: [x^0] W_n = 2 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 = 2: [x^0] W_n = 2 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 >= 2, with A_n(x) = (2) + (2) x, B_n(x) = (2) + (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 >= 1, 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 >= 2","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":1,"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 >= 2, with A_n(x) = (2) + (2) x, B_n(x) = (2) + (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 >= 1, 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 >= 2"],"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"}