{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["6","8","1"]],"recurrence":{"denominator":{"coeffs":[["12"],["14"],["4"]]},"terms":[{"coeff":{"coeffs":[["35","25"],["46","35"],["15","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","11","-13"],["1","7","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-84","-1008","-924"],["-156","-1522","-1286"],["-84","-829","-659"],["-14","-190","-147"],["0","-15","-12"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-12","-168","-1476","-360"],["-10","-258","-1940","-756"],["-2","-116","-957","-497"],["0","-16","-204","-131"],["0","0","-15","-12"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-6","-42","-234","282"],["0","-5","-51","-209","265"],["0","-1","-16","-64","81"],["0","0","-1","-7","8"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["30","270"],["52","378"],["26","198"],["4","46"],["0","4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","30","270"],["0","52","378"],["0","26","198"],["0","4","46"],["0","0","4"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["6","16","3"]],"recurrence":{"denominator":{"coeffs":[["0","48"],["12","104"],["14","84"],["4","30"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["-2","140","162"],["30","374","361"],["42","349","290"],["14","130","99"],["0","15","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","6","82","-90"],["3","22","102","-127"],["1","13","43","-57"],["0","1","7","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["30","22","4"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["6","8","1"]],"recurrence":{"denominator":{"coeffs":[["12"],["14"],["4"]]},"terms":[{"coeff":{"coeffs":[["35","25"],["46","35"],["15","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","11","-13"],["1","7","-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":[["0","-30","-240","270"],["0","-52","-326","378"],["0","-26","-172","198"],["0","-4","-42","46"],["0","0","-4","4"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-30","-240","270"],["0","0","-52","-326","378"],["0","0","-26","-172","198"],["0","0","-4","-42","46"],["0","0","0","-4","4"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["4"],["4"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-3","-3"],["-1","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","-1","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","-90","90","-2790","2790"],["0","-96","96","-4056","4056"],["0","-34","34","-2206","2206"],["0","-4","4","-532","532"],["0","0","0","-48","48"]]},"deriv_order":0}],"row_multiplier":["0","-1","1"],"start_base":0,"step_multiplier":{"coeffs":[["0","0","1","-2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","144","-216","0","72"],["0","0","94","-129","-24","59"],["0","0","15","-18","-9","12"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","3","12","-54","60","-21"],["0","0","1","5","-21","23","-8"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-96","-72"],["-76","-59"],["-15","-12"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","-18","21"],["-1","-7","8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["30"],["22"],["4"]]},"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 + (1) theta; 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, 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 + (1) theta","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["6","8","1"]],"recurrence":{"denominator":{"coeffs":[["12"],["14"],["4"]]},"terms":[{"coeff":{"coeffs":[["35","25"],["46","35"],["15","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","11","-13"],["1","7","-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 + (1) theta, for every n >= 0"},"parent_proof":{"route":"hypergeometric_row","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":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = ((15/4 n^2 + 23/2 n + 35/4)/(n^2 + 7/2 n + 3)) + ((3 n^2 + 35/4 n + 25/4)/(n^2 + 7/2 n + 3)) x and b_n(x) = ((1/4 n + 1/2)/(n^2 + 7/2 n + 3)) + ((7/4 n + 11/4)/(n^2 + 7/2 n + 3)) x + ((-2 n - 13/4)/(n^2 + 7/2 n + 3)) x^2; open hypotheses: derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = n + 2 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 = 0 (not a disproof): fails at n = 1: [x^0] b_n = 1/10 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): c_1 is not gamma(n) w(x)","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: P_n = a_n(x) S_n(P_(n-1)) for n >= 1, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 35 + 46 n + 15 n^2 + 25 x + 35 n x + 12 n^2 x, B = 2 + n + 11 x + 7 n x - 13 x^2 - 8 n x^2, C = 0) / (4 n^2 + 14 n + 12); open hypotheses: weyl_step_leading_coefficients_same_sign: fails at n = 1: A_1 B_lead = -96 n^3 - 436 n^2 - 655 n - 325 = -1512 there (exact); weyl_step_interlacing: fails at n = 1: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = 23328 n^6 + 216756 n^5 + 838188 n^4 + 1726569 n^3 + 1998054 n^2 + 1231605 n + 315900 = 6350400 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 1: A_a B_(a+1) = -96 n^3 - 436 n^2 - 655 n - 325 = -1512 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 0, with p = (-1) (k) (k + n + 2), q = (1) (k - n - 3) (k - n - 3); valuation v(n) = 0, degree d(n) = n + 2: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0): for n >= 0 the coefficients U_j (U_j = [x^(j + v(n))] P_n, v(n) = 0) satisfy U_j = U_0(n) prod_(t <= j) R(n, t), R(n, j) = (1) (j - n - 3) (j - n - 3) / ((j) (j + n + 2)), so sum_j U_j y^j is U_0(n) times the image of (1 + y)^(n + 2) under a nonnegative multiplier sequence, and U_0(n) != 0 for n >= 0 by its scalar recurrence","route":"hypergeometric_row","success":true,"variant":"hypergeometric_row"}],"conditional_on":[],"route":"hypergeometric_row","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 + (1) theta 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"}