{"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":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows, A007318 rows repeated three times."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["42","171","129","-27","-27"],["34","-2","-63","-27","0"],["-15","-24","-9","0","0"],["-1","-1","0","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-39","36","102","0","-27"],["61","52","-36","-27","0"],["-15","-24","-9","0","0"],["-1","-1","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-108","-216","-108"],["27","54","27"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-27","-81","-81","-27"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["-21","-75","-27","27"],["-16","9","27","0"],["6","9","0","0"],["1","0","0","0"]]},"deriv_order":0,"shift":3},{"coeff":{"coeffs":[["-21","-75","-27","27"],["-16","9","27","0"],["6","9","0","0"],["1","0","0","0"]]},"deriv_order":1,"shift":3}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["54","-21","-108","27"],["-25","-45","27","0"],["-3","9","0","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["168","12","-237","-54","27"],["-97","-115","9","27","0"],["6","15","9","0","0"],["1","1","0","0","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["162","297","108","-27"],["-27","-54","-27","0"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["27","81","81","27"]]},"deriv_order":2,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","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":[["-567","-3726","-8505","-8100","-2025","1458","729"],["-432","-1053","162","2484","2430","729","0"],["162","729","1215","891","243","0","0"],["27","81","81","27","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["-567","-3726","-8505","-8100","-2025","1458","729"],["-432","-1053","162","2484","2430","729","0"],["162","729","1215","891","243","0","0"],["27","81","81","27","0","0","0"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-2","0"],["0","-3"],["0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","18","18"],["0","-6","-6"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","9","18","9"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"shift":2}]},"quotient":[],"step_multiplier":{"coeffs":[["5832","34992","87480","116640","87480","34992","5832"]]},"step_quotients":[[],[{"coeff":{"coeffs":[["-5832","-40824","-122472","-204120","-204120","-122472","-40824","-5832"],["0","-2916","-17496","-43740","-58320","-43740","-17496","-2916"],["0","2916","17496","43740","58320","43740","17496","2916"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","34992","244944","734832","1224720","1224720","734832","244944","34992"],["0","-17496","-122472","-367416","-612360","-612360","-367416","-122472","-17496"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","26244","209952","734832","1469664","1837080","1469664","734832","209952","26244"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","17496","119556","349920","568620","554040","323676","104976","14580"],["0","0","-2916","-17496","-43740","-58320","-43740","-17496","-2916"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","8748","61236","183708","306180","306180","183708","61236","8748"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["-2","0"],["0","1"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-12","-12"],["0","6","6"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-9","-18","-9"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"shift":1}]},"quotient":[],"step_multiplier":{"coeffs":[["11664","69984","174960","233280","174960","69984","11664"]]},"step_quotients":[[{"coeff":{"coeffs":[["11664","69984","174960","233280","174960","69984","11664"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["-11664","-69984","-174960","-233280","-174960","-69984","-11664","0"],["0","-5832","-34992","-87480","-116640","-87480","-34992","-5832"],["0","-5832","-34992","-87480","-116640","-87480","-34992","-5832"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-34992","-244944","-734832","-1224720","-1224720","-734832","-244944","-34992"],["0","34992","244944","734832","1224720","1224720","734832","244944","34992"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","-52488","-419904","-1469664","-2939328","-3674160","-2939328","-1469664","-419904","-52488"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","-34992","-227448","-629856","-962280","-874800","-472392","-139968","-17496"],["0","0","5832","34992","87480","116640","87480","34992","5832"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-17496","-122472","-367416","-612360","-612360","-367416","-122472","-17496"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-2"],["3"],["-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["60","60"],["-45","-45"],["9","9"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["108","216","108"],["-27","-54","-27"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["27","81","81","27"]]},"deriv_order":3,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["-27","-81","-81","-27"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["314928","2834352","11337408","26453952","39680928","39680928","26453952","11337408","2834352","314928"]]},"step_quotients":[[],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["157464","1417176","5668704","13226976","19840464","19840464","13226976","5668704","1417176","157464"],["0","0","0","0","0","0","0","0","0","0"],["-157464","-1417176","-5668704","-13226976","-19840464","-19840464","-13226976","-5668704","-1417176","-157464"]]},"deriv_order":0},{"coeff":{"coeffs":[["3779136","37791360","170061120","453496320","793618560","952342272","793618560","453496320","170061120","37791360","3779136"],["-4251528","-42515280","-191318760","-510183360","-892820880","-1071385056","-892820880","-510183360","-191318760","-42515280","-4251528"],["1417176","14171760","63772920","170061120","297606960","357128352","297606960","170061120","63772920","14171760","1417176"]]},"deriv_order":1},{"coeff":{"coeffs":[["12754584","140300424","701502120","2104506360","4209012720","5892617808","5892617808","4209012720","2104506360","701502120","140300424","12754584"],["-4251528","-46766808","-233834040","-701502120","-1403004240","-1964205936","-1964205936","-1403004240","-701502120","-233834040","-46766808","-4251528"]]},"deriv_order":2},{"coeff":{"coeffs":[["4251528","51018336","280600848","935336160","2104506360","3367210176","3928411872","3367210176","2104506360","935336160","280600848","51018336","4251528"]]},"deriv_order":3}],[{"coeff":{"coeffs":[["1732104","17478504","79361856","213521184","376968816","456330672","383582304","221079456","83613384","18738216","1889568"],["-1417176","-13856832","-60938568","-158723712","-271153008","-317447424","-257926032","-143607168","-52435512","-11337408","-1102248"],["0","157464","1417176","5668704","13226976","19840464","19840464","13226976","5668704","1417176","157464"]]},"deriv_order":0},{"coeff":{"coeffs":[["4251528","47239200","238557960","722759760","1459691280","2063408256","2083248720","1502206560","758189160","255091680","51490728","4723920"],["0","-944784","-9447840","-42515280","-113374080","-198404640","-238085568","-198404640","-113374080","-42515280","-9447840","-944784"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","1417176","15588936","77944680","233834040","467668080","654735312","654735312","467668080","233834040","77944680","15588936","1417176"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["27","81","81","27"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":0,"shift":0},{"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":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","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":{"route":"periodic_product","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 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":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_proof":{"attempts":[{"detail":"c and the seeds are nonzero, real-rooted and of one sign, so P_(n0 + k m + i) = c^m P_(n0 + i) is too","route":"periodic_product","success":true,"variant":"P_(n+k) = c(x) P_n"}],"conditional_on":[],"route":"periodic_product","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 = phi(n, D) = 1 + (1) D 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"}
