{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","3","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-24","-98","-159","-130","-56","-12","-1"],["-26","-96","-135","-89","-27","-3","0"],["-9","-30","-36","-18","-3","0","0"],["-1","-3","-3","-1","0","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-24","-98","-159","-130","-56","-12","-1"],["-26","-96","-135","-89","-27","-3","0"],["-9","-30","-36","-18","-3","0","0"],["-1","-3","-3","-1","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["36","129","174","108","30","3"],["57","162","162","66","9","0"],["24","57","42","9","0","0"],["3","6","3","0","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["36","129","174","108","30","3"],["57","162","162","66","9","0"],["24","57","42","9","0","0"],["3","6","3","0","0","0"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["-24","-66","-63","-24","-3"],["-42","-84","-51","-9","0"],["-21","-30","-9","0","0"],["-3","-3","0","0","0"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["-24","-66","-63","-24","-3"],["-42","-84","-51","-9","0"],["-21","-30","-9","0","0"],["-3","-3","0","0","0"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["6","11","6","1"],["11","12","3","0"],["6","3","0","0"],["1","0","0","0"]]},"deriv_order":0,"shift":3},{"coeff":{"coeffs":[["6","11","6","1"],["11","12","3","0"],["6","3","0","0"],["1","0","0","0"]]},"deriv_order":1,"shift":3}]},"exceptional_bases":[0,1],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["0"],["3","4","1"]],"recurrence":{"denominator":{"coeffs":[["0","2","-3","1"],["2","-6","3","0"],["-3","3","0","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-1","-3","-2","2","3","1"],["-1","-3","0","8","9","3","0"],["0","3","9","9","3","0","0"],["1","3","3","1","0","0","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","6","15","9","-3","-3"],["6","18","9","-12","-9","0"],["3","-3","-15","-9","0","0"],["-3","-6","-3","0","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","3","-9","-3","3"],["-3","-15","-3","9","0"],["-6","3","9","0","0"],["3","3","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["0","0","1","-2","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","3","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":3,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["6","17","17","7","1"],["-1","-11","-19","-11","-2"],["-10","-20","-10","1","1"],["0","6","9","3","0"],["4","7","3","0","0"],["1","1","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["6","17","17","7","1"],["-1","-11","-19","-11","-2"],["-10","-20","-10","1","1"],["0","6","9","3","0"],["4","7","3","0","0"],["1","1","0","0","0"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-1","-2","-1"],["-1","-2","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","-3"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"shift":2}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0"],["12","30","24","6"],["0","-15","-24","-9"],["-21","-39","-18","0"],["-3","6","12","3"],["9","15","6","0"],["3","3","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["-6","-21","-27","-15","-3"],["3","18","33","24","6"],["9","21","12","-3","-3"],["-3","-12","-15","-6","0"],["-3","-6","-3","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0"],["-1"],["1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0"],["-3","-3"],["3","3"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","0"],["4","8","4"],["-4","-8","-4"]]},"deriv_order":0}],[]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0"],["-18","-63","-81","-45","-9"],["-9","-12","9","18","6"],["33","105","120","57","9"],["15","24","-3","-18","-6"],["-15","-42","-39","-12","0"],["-6","-12","-6","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["18","81","144","126","54","9"],["-9","-60","-141","-153","-78","-15"],["-24","-78","-84","-24","12","6"],["9","39","63","45","12","0"],["6","18","18","6","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0"],["-1"],["1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0"],["-1"],["1"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0"],["1","1"],["-1","-1"]]},"deriv_order":0}],[]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["12","54","96","84","36","6"],["-12","-54","-96","-84","-36","-6"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["1","1"],["-2","-2"],["1","1"]]},"step_quotients":[[],[{"coeff":{"coeffs":[["1","1"],["0","0"],["-1","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1","-2","-1"],["1","2","1"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-1","-2","-1"],["0","0","0"],["1","2","1"]]},"deriv_order":0}]]}],"restart_bases":[0,1],"row_multiplier":{"coeffs":[["1","1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-3","-3","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["3","6","3"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-3","-3"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":3}]},"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 = phi(n, D) = 1 + (1) D; 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.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"derivative_polynomial_real_rooted","statement":"the polynomial in D is real-rooted 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":"phi(n, D) = 1 + (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","3","3","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D, for every n >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"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 differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n"},"verdict":"open"},"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":"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: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","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":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a derivative or a lag term","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":"the parent equals the flagged entry, but the flag gives real-rootedness only and one-signed coefficients are not shown: the weight of the coefficient of x^s of P_(n-2) in the coefficient of x^(s+0) is not shown to be >= 0 for n >= 3","route":"attestation","success":false,"variant":"oeis_catalog:A237765:lean_proved"},{"detail":"no lean_proved catalog entry equals the parent; 1 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A172107:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D for every n >= 0 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}
