{"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"]],"recurrence":{"denominator":{"coeffs":[["0"],["-2"],["4"]]},"terms":[{"coeff":{"coeffs":[["6","6","0"],["-27","-26","-1"],["27","24","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","3","0"],["0","-9","-10","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-12","-22","-8","0"],["-72","-157","-74","-3"],["-135","-348","-189","-8"],["-81","-279","-177","-7"],["0","-54","-48","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","12","40","36","6"],["0","0","43","70","13"],["0","-81","-162","-43","6"],["0","-81","-243","-137","-3"],["0","0","-54","-48","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","12","31","22","3"],["0","0","36","100","72","8"],["0","0","27","93","73","7"],["0","0","0","18","20","2"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["4","2"],["18","12"],["26","26"],["12","24"],["0","8"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","4","2"],["0","18","12"],["0","26","26"],["0","12","24"],["0","0","8"]]},"deriv_order":1,"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"],["2","0"],["-10","2"],["12","-8"],["0","8"]]},"terms":[{"coeff":{"coeffs":[["-6","-12","-6","0"],["45","82","35","0"],["-108","-165","-54","-1"],["81","63","-15","-1"],["0","54","48","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-6","-3","0"],["0","18","32","14","0"],["0","-27","-39","-13","-1"],["0","0","-18","-20","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["2","6","4"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-2"],["4"]]},"terms":[{"coeff":{"coeffs":[["6","6","0"],["-27","-26","-1"],["27","24","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","3","0"],["0","-9","-10","-1"]]},"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","4","6","2"],["0","18","30","12"],["0","26","52","26"],["0","12","36","24"],["0","0","8","8"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","4","6","2"],["0","0","18","30","12"],["0","0","26","52","26"],["0","0","12","36","24"],["0","0","0","8","8"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["0"],["2"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1"],["-3","-3"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","0","22","26","4"],["0","0","0","130","150","20"],["0","0","0","284","320","36"],["0","0","0","272","300","28"],["0","0","0","96","104","8"]]},"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","18","46","37","8","-1"],["0","0","45","121","107","31","0"],["0","0","27","78","76","26","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-6","-19","-21","-9","-1"],["0","0","0","-9","-28","-30","-12","-1"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-6","-4","0"],["-27","-22","-1"],["-27","-24","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","6","7","1"],["0","9","10","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2"],["6"],["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":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-2"],["4"]]},"terms":[{"coeff":{"coeffs":[["6","6","0"],["-27","-26","-1"],["27","24","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","3","0"],["0","-9","-10","-1"]]},"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":"Unresolved: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = ((27/4 n^2 - 27/4 n + 3/2)/(n^2 - 1/2 n)) + ((6 n^2 - 13/2 n + 3/2)/(n^2 - 1/2 n)) x + ((1/4 n^2 - 1/4 n)/(n^2 - 1/2 n)) x^2 and b_n(x) = ((-9/4 n + 3/4)/(n^2 - 1/2 n)) x + ((-5/2 n + 3/4)/(n^2 - 1/2 n)) x^2 + ((-1/4 n)/(n^2 - 1/2 n)) x^3; 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 = 1 (not a disproof): fails at n = 1: [x^1] b_n = -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":"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): deg_x c_0 > 1 or deg_x c_1 != 2","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: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = 2 n - 2 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 1, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = n - n^2 + 4 n^2 x - 3 n x^2 + 9 n^2 x^2, B = n x - 3 x^2 + 10 n x^2 - 3 x^3 + 9 n x^3, C = 0) / (4 n^2 - 2 n); open hypotheses: weyl_step_interlacing: at n = 1: A does not strictly interlace B (exact Bezout test)","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 = (1/2) (k - 3 n - 1) (k), q = (1/2) (k - 2 n - 1) (k - n - 1); valuation v(n) = 0, degree d(n) = n: 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^(d(n) - j)] P_n, d(n) = n) satisfy U_j = U_0(n) prod_(t <= j) R(n, t), R(n, j) = (-1) (j + 2 n) (j - n - 1) / ((j + n) (j)), so sum_j U_j y^j is U_0(n) times the image of (1 + y)^(n) 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"}