{"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"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","4"],["2","6"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","2"],["0","2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"theta_plus_one of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-3","-27","324","-192"],["-4","24","564","-544"],["-1","59","329","-640"],["0","28","44","-400"],["0","4","-26","-140"],["0","0","-10","-26"],["0","0","-1","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-25","159","972","-832"],["0","-20","448","1436","-2272"],["0","-3","369","539","-2560"],["0","0","116","-156","-1520"],["0","0","12","-158","-500"],["0","0","0","-38","-86"],["0","0","0","-3","-6"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-23","273","324","-512"],["0","0","-14","494","244","-1344"],["0","0","-1","299","-231","-1440"],["0","0","0","68","-316","-800"],["0","0","0","4","-126","-240"],["0","0","0","0","-20","-36"],["0","0","0","0","-1","-2"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","-4","60","0","-64"],["0","0","0","-2","94","-64","-160"],["0","0","0","0","48","-112","-160"],["0","0","0","0","8","-72","-80"],["0","0","0","0","0","-20","-20"],["0","0","0","0","0","-2","-2"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["3","-21","12"],["4","-64","52"],["1","-67","91"],["0","-28","82"],["0","-4","40"],["0","0","10"],["0","0","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","9","-63","36"],["0","12","-192","156"],["0","3","-201","273"],["0","0","-84","246"],["0","0","-12","120"],["0","0","0","30"],["0","0","0","3"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","3","-21","12"],["0","0","4","-64","52"],["0","0","1","-67","91"],["0","0","0","-28","82"],["0","0","0","-4","40"],["0","0","0","0","10"],["0","0","0","0","1"]]},"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"],["2","2","0"],["1","-7","0"],["0","-12","2"],["0","-4","5"],["0","0","4"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["0","16","-28","4"],["2","26","-98","22"],["1","1","-126","50"],["0","-12","-68","60"],["0","-4","-9","40"],["0","0","4","14"],["0","0","1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-6","-6","2"],["0","2","-22","-14","10"],["0","0","-24","-4","20"],["0","0","-8","12","20"],["0","0","0","10","10"],["0","0","0","2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["3","4","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"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","4"],["2","6"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","2"],["0","2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","3","-15","-27","3","12"],["0","0","4","-56","-72","40","52"],["0","0","1","-65","-42","115","91"],["0","0","0","-28","26","136","82"],["0","0","0","-4","32","76","40"],["0","0","0","0","10","20","10"],["0","0","0","0","1","2","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","9","-45","-81","9","36"],["0","0","0","12","-168","-216","120","156"],["0","0","0","3","-195","-126","345","273"],["0","0","0","0","-84","78","408","246"],["0","0","0","0","-12","96","228","120"],["0","0","0","0","0","30","60","30"],["0","0","0","0","0","3","6","3"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","3","-15","-27","3","12"],["0","0","0","0","4","-56","-72","40","52"],["0","0","0","0","1","-65","-42","115","91"],["0","0","0","0","0","-28","26","136","82"],["0","0","0","0","0","-4","32","76","40"],["0","0","0","0","0","0","10","20","10"],["0","0","0","0","0","0","1","2","1"]]},"deriv_order":2}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["-3"],["-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","0","0","240","288","-144","-192"],["0","0","0","0","548","408","-828","-688"],["0","0","0","0","438","-124","-1562","-1000"],["0","0","0","0","148","-460","-1364","-756"],["0","0","0","0","18","-278","-610","-314"],["0","0","0","0","0","-68","-136","-68"],["0","0","0","0","0","-6","-12","-6"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","108","72","-180","-144"],["0","0","0","0","0","246","-12","-762","-504"],["0","0","0","0","0","196","-324","-1236","-716"],["0","0","0","0","0","66","-398","-994","-530"],["0","0","0","0","0","8","-200","-424","-216"],["0","0","0","0","0","0","-46","-92","-46"],["0","0","0","0","0","0","-4","-8","-4"]]},"deriv_order":1}],"row_multiplier":["0","0","1","2","1"],"start_base":0,"step_multiplier":{"coeffs":[["0","0","1","2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","7","34","47","20"],["0","0","6","26","34","14"],["0","0","1","4","5","2"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","4","12","12","4"],["0","0","0","2","6","6","2"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-3","-12"],["-4","-10"],["-1","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-4","-4"],["0","-2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3"],["4"],["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"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","4"],["2","6"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","2"],["0","2","2"]]},"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":{"route":"ma_wang_interval","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: the theorem needs at least two seed rows","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"Unresolved: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = ((n^2 + 2 n)/(n^2 + 2 n)) + ((2 n^2 + 6 n + 4)/(n^2 + 2 n)) x and b_n(x) = ((0)/(n^2 + 2 n)) + ((2 n + 2)/(n^2 + 2 n)) x + ((2 n + 2)/(n^2 + 2 n)) x^2; open hypotheses: 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 = 2 (not a disproof): fails at n = 1: [x^2] b_n = 4/3 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":"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 = 0, alpha_n = (2 n^2 + 6 n + 4)/(n^2 + 2 n), beta_n = (n^2 + 2 n)/(n^2 + 2 n), kappa_n = (2 n + 2)/(n^2 + 2 n): every P_n, n >= 0, is nonzero and real-rooted with every root in (-1, 0); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 1","route":"ma_wang_interval","success":true,"variant":"ma_wang_interval"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: 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 = 0, alpha_n = (2 n^2 + 6 n + 4)/(n^2 + 2 n), beta_n = (n^2 + 2 n)/(n^2 + 2 n), kappa_n = (2 n + 2)/(n^2 + 2 n)","every P_n, n >= 0, is nonzero and real-rooted with every root in (-1, 0); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 1"],"theorem":"ma_wang_interval: S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"ma_wang_interval","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"}