{"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":[["1"],["1","1"],["0","2","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle A127647 * A097806, read by rows."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","-14"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","-24"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0","-5"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","-6"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","-16"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","-5"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["0","-8"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","0","2"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["0","0","0","5"]]},"deriv_order":2,"shift":2},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"shift":3},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"shift":3}]},"exceptional_bases":[0],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1","2"],["0","4","6"]],"recurrence":{"denominator":{"coeffs":[["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","14"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","5"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","5"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["0","1","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","1"],["0","2","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","-4"],["0","-4"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","4"],["0","0","4"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","5"],["0","0","0","5"]]},"deriv_order":2}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0"],["2"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0"],["0","2"]]},"deriv_order":1}],"shift":1}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0"],["0","-1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0"],["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 1) 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 3 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":[["1"],["1","1"],["0","2","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","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 + (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":[],"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":"HypothesisFailed: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 3, with a_n = (0)/(n - 2), b_n = (n - 2)/(n - 2), c_n = (-n + 2)/(n - 2); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 3: f(n) a_n = 0 = 0 there (exact); seeds_interlace: deg Q_2 != deg Q_1 + 1 = 2; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; recurrence_coefficient_positive: fails at n = 4: the denominator of beta_n = -n^2 + 5 n - 6 vanishes there; root_upper_bound: not attempted: the real-rootedness hypotheses do not all hold","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 >= 3, with A_n(x) = ((n - 2)/(n - 2)) x, V_n(x) = 0, W_n(x) = ((n - 2)/(n - 2)) x^2; open hypotheses: seed_roots_negative: P_2 is zero, not real-rooted, or has a root >= 0 (exact); seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); constant_term_positive: fails at n = 3: A_n(0) = (0)/(n - 2) = 0 there (exact); lag_coefficient_nonpositive: W_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2, which is negative on (-inf, 0) (no root there, exact): fails at n = 3: gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) = -1/2 there (exact); derivative_or_lag_coefficient_negative: W_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2, which is negative on (-inf, 0) (no root there, exact): fails at n = 3: gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) = -1/2 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":"HypothesisFailed: P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 3, with A_n(x) = ((n - 2)/(n - 2)) x, B_n(x) = ((n - 2)/(n - 2)) x^2; open hypotheses: lag_coefficient_nonpositive: neither everywhere (the quadratic test p_2 <= 0, p_0 <= 0, p_1^2 <= 4 p_2 p_0 does not hold for every n (not a disproof): fails at n = 3: p_2 = [x^2] B_n = 1 there (exact); p_0 = [x^0] B_n <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of p_0 = [x^0] B_n at n = 3 + u have one sign); p_1^2 - 4 p_2 p_0 <= 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of p_1^2 - 4 p_2 p_0 at n = 3 + u have one sign)) nor on x <= 0 (B_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2, which is negative on (-inf, 0) (no root there, exact): fails at n = 3: gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) = -1/2 there (exact))","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 1, with p = (-1) (k - n + 1), q = (-1) (k - n - 1); valuation v(n) = n - 1, degree d(n) = n: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]: for n >= 1 the coefficients U_j (U_j = [x^(j + v(n))] P_n, v(n) = n - 1) satisfy U_j = U_0(n) prod_(t <= j) R(n, t), R(n, j) = (-1) (j - 2) / ((j)), so sum_j U_j y^j is U_0(n) times the image of (1 + y)^(1) under a nonnegative multiplier sequence, and U_0(n) != 0 for n >= 1 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"}