{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013611 of Riordan array (1, 2+2x)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-4","-4","36","-44","16"],["4","-12","12","-4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","-7","23","3","-32","16"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2","-6","6","-2"],["2","-6","6","-2"]]},"deriv_order":0,"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","4","2"]]},"terms":[{"coeff":{"coeffs":[["11","68","138","116","35"],["-7","-40","-94","-96","-35"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","31","110","174","127","35"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1","1"],"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-4","-16"],["4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","-16","-16"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2"],["2"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["2","-6","6","-2"],["2","-6","6","-2"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["12","112","320","256"],["24","224","640","512"],["12","112","320","256"]]},"step_quotients":[[{"coeff":{"coeffs":[["48","544","2176","3584","2048"],["96","1088","4352","7168","4096"],["48","544","2176","3584","2048"]]},"deriv_order":0},{"coeff":{"coeffs":[["-18","-264","-1472","-3840","-4608","-2048"],["-18","-264","-1472","-3840","-4608","-2048"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-18","-192","-704","-1024","-512"],["-18","-192","-704","-1024","-512"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-8","-64"],["8","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["12","80","128"],["-12","-64","-64"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3","28","80","64"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["72","1344","10112","38912","79872","81920","32768"],["216","4032","30336","116736","239616","245760","98304"],["216","4032","30336","116736","239616","245760","98304"],["72","1344","10112","38912","79872","81920","32768"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","-2304","-43008","-323584","-1245184","-2555904","-2621440","-1048576"],["288","-1536","-88576","-815104","-3416064","-7340032","-7733248","-3145728"],["864","9216","-7680","-503808","-2777088","-6684672","-7471104","-3145728"],["864","13824","78336","143360","-286720","-1572864","-2228224","-1048576"],["288","5376","40448","155648","319488","327680","131072","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","576","13056","123904","634880","1884160","3211264","2883584","1048576"],["-432","-9216","-79872","-352256","-778240","-524288","1048576","2097152","1048576"],["-864","-20160","-198912","-1076224","-3461120","-6701056","-7536640","-4456448","-1048576"],["-432","-10368","-105984","-600064","-2048000","-4292608","-5373952","-3670016","-1048576"]]},"deriv_order":1},{"coeff":{"coeffs":[["108","3024","36864","256000","1112064","3121152","5636096","6291456","3932160","1048576"],["216","6048","73728","512000","2224128","6242304","11272192","12582912","7864320","2097152"],["108","3024","36864","256000","1112064","3121152","5636096","6291456","3932160","1048576"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["-144","-2112","-9472","3072","151552","475136","589824","262144"],["-432","-6912","-39168","-71680","143360","786432","1114112","524288"],["-432","-7488","-49920","-152576","-167936","147456","458752","262144"],["-144","-2688","-20224","-77824","-159744","-163840","-65536","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["108","2592","26496","150016","512000","1073152","1343488","917504","262144"],["216","5184","52992","300032","1024000","2146304","2686976","1835008","524288"],["108","2592","26496","150016","512000","1073152","1343488","917504","262144"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["2"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-2","-8"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-8"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["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 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0); 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":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.splits_fPolynomial"],"lean_route":"fTransform","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":0,"input_interlacing":{"reason":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0)","kind":"preserver_image","operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"mobius"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0), for every n >= 0"},"parent_proof":{"route":"liu_wang_three_term","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 [-1, 0]","input_root_region":{"lower":{"closed":true,"value":"-1"},"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":"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":["the division remainder is removed by row annihilators of the shifted parent"],"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","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: coefficient of P_(n-1) must be 1 after normalization; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","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 >= 2, with A_n(x) = (2) + (8) x, V_n(x) = 0, W_n(x) = (2) + (8) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = 2 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 2 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 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":"P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 2, with A_n(x) = (2) + (8) x, B_n(x) = (2) + (8) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/4] (so its coefficients have one sign); for n >= 1, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 2","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 2, with A_n(x) = (2) + (8) x, B_n(x) = (2) + (8) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/4] (so its coefficients have one sign); for n >= 1, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 2"],"theorem":"liu_wang_three_term: L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"liu_wang_three_term","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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0) 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 [-1, 0]"},"kind":"preserver_image"}