{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","2"],["-11/2","-8"],["15/2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1/2","-1/2","1"],["1/2","7/2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-6","0"],["-19","-8"],["-15","-12"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-6","6"],["-1","-7","8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["6"],["10"],["4"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["4"]]},"terms":[{"coeff":{"coeffs":[["6","4"],["-27","-16"],["27","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","2"],["0","-9","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["3","5","2"],"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","2"],["-11/2","-8"],["15/2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1/2","-1/2","1"],["1/2","7/2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["6"],["10"],["4"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-3","0"],["-19/2","-4"],["-15/2","-6"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-3","3"],["-1/2","-7/2","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3"],["5"],["2"]]},"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 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((1) x + (1)); so the candidate rows satisfy the input rule at every base n >= 1 (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 >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"lift_summary":{"hypotheses":[{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":1,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (1))","kind":"preserver_image","operator":{"intercept":"1","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","2"],["-11/2","-8"],["15/2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1/2","-1/2","1"],["1/2","7/2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)), for every n >= 1"},"parent_proof":{"route":"hypergeometric_row","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, -1]","input_root_region":{"upper":{"closed":true,"value":"-1"}},"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":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically"},"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":"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 >= 2, with a_n(x) = ((15/4 n^2 - 11/4 n + 1/2)/(n^2 + 1/2 n)) + ((3 n^2 - 4 n + 1)/(n^2 + 1/2 n)) x and b_n(x) = ((1/4 n - 1/4)/(n^2 + 1/2 n)) + ((7/4 n - 1/4)/(n^2 + 1/2 n)) x + ((-2 n + 1/2)/(n^2 + 1/2 n)) x^2; open hypotheses: derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = 1/2 n - 1/2 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 = 2: [x^0] b_n = 1/20 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 >= 2, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 1 - 11/2 n + 15/2 n^2 + 2 x - 8 n x + 6 n^2 x, B = -1/2 + 1/2 n - 1/2 x + 7/2 n x + x^2 - 4 n x^2, C = 0) / (2 n^2 + n); open hypotheses: weyl_step_leading_coefficients_same_sign: fails at n = 2: A_1 B_lead = -24 n^3 + 38 n^2 - 16 n + 2 = -70 there (exact); weyl_step_interlacing: fails at n = 2: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = 1458 n^6 - 6075/2 n^5 + 9801/4 n^4 - 1917/2 n^3 + 729/4 n^2 - 27/2 n = 28350 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 2: A_a B_(a+1) = -24 n^3 + 38 n^2 - 16 n + 2 = -70 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 >= 1, with p = (-1) (k) (k + n + 1), q = (1) (k - n - 1) (k - n); valuation v(n) = 0, degree d(n) = n - 1: every P_n, n >= 1, 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) = 0) satisfy U_j = U_0(n) prod_(t <= j) R(n, t), R(n, j) = (1) (j - n - 1) (j - n) / ((j) (j + n + 1)), so sum_j U_j y^j is U_0(n) times the image of (1 + y)^(n - 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 >= 1, 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 = P((1) x + (1)) for every n >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, -1]"},"kind":"preserver_image"}