{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013610 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["264","1656","2646","-405","-1944","-729"],["-156","-1368","-2997","-1377","0","0"],["-684","-1890","81","729","0","0"],["216","486","0","0","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["296","1824","2394","-3348","-7695","-4374","-729"],["-204","-2016","-5859","-6480","-3402","-729","0"],["-828","-2754","-1377","243","0","0","0"],["216","486","0","0","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["32","168","-252","-2943","-5751","-3645","-729"],["-48","-648","-2862","-5103","-3402","-729","0"],["-144","-864","-1458","-486","0","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["-36","-36","-459","0","243"],["108","54","405","243","0"],["0","162","0","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0"],["-36","-36","-459","0","243"],["108","54","405","243","0"],["0","162","0","0","0"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["144","-72","864","243","-243"],["-252","342","-1269","-486","243"],["108","-432","405","243","0"],["0","162","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["480","-648","-5724","-1701","1944","729"],["-1860","-3870","3159","2835","0","0"],["1332","3348","-81","-729","0","0"],["-216","-486","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["64","48","-1152","-1674","2349","2916","729"],["-240","-1080","-54","4131","3402","729","0"],["144","864","1458","486","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","0","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-45","-54"],["45","27"],["-18","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-60","-207","-108"],["24","72","27"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-8","-48","-81","-27"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["9"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","-20736","-269568","-1679616","-6998400","-20356596","-36610380","-31591215","-354294","17006112","9565938","1594323"],["0","0","62208","777600","4105728","12667104","27687420","46491246","54147933","36315135","12223143","1594323","0"],["0","0","0","93312","1119744","5248800","11967264","13345074","6377292","1062882","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","-20736","-269568","-1679616","-6998400","-20356596","-36610380","-31591215","-354294","17006112","9565938","1594323"],["0","0","62208","777600","4105728","12667104","27687420","46491246","54147933","36315135","12223143","1594323","0"],["0","0","0","93312","1119744","5248800","11967264","13345074","6377292","1062882","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","373248","6718464","51648192","220449600","568287576","901323936","859340097","463947993","129140163","14348907"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","1119744","22394880","195255360","971237952","3027560328","6113697264","7985963907","6547884561","3171108447","817887699","86093442"],["0","0","0","-746496","-13436928","-103296384","-440899200","-1136575152","-1802647872","-1718680194","-927895986","-258280326","-28697814","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","-1492992","-32472576","-310728960","-1716987456","-6044728032","-14113655784","-22071807612","-22857808851","-15209841420","-6170030010","-1377495072","-129140163"],["0","0","0","995328","20901888","192595968","1021206528","3433974912","7611179904","11207027808","10815887232","6633978003","2463229035","502211745","43046721"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","-331776","-7962624","-85100544","-532998144","-2165864832","-5971034880","-11346855840","-14812323552","-13027213233","-7455054348","-2630632950","-516560652","-43046721"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","-497664","-9704448","-81181440","-377073792","-1043671392","-1676991600","-1243571940","366694290","1477937421","1114431777","358722675","43046721"],["0","0","-746496","-13436928","-103296384","-440899200","-1136575152","-1802647872","-1718680194","-927895986","-258280326","-28697814","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","331776","6469632","52627968","222269184","448877376","-73693152","-2766799944","-7259484060","-9830595618","-7754787072","-3539397060","-860934420","-86093442"],["0","0","995328","20901888","192595968","1021206528","3433974912","7611179904","11207027808","10815887232","6633978003","2463229035","502211745","43046721"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","-331776","-7962624","-85100544","-532998144","-2165864832","-5971034880","-11346855840","-14812323552","-13027213233","-7455054348","-2630632950","-516560652","-43046721"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["-12","0","27"],["12","-18","-27"],["0","18","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-8","12","126","81"],["0","-24","-72","-27"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","8","48","81","27"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","-36288","-979776","-8713008","-36811584","-81192375","-90404019","-39680928","4251528","7971615","1594323"],["0","31104","171072","-583200","-6123600","-14552298","-6948099","17832798","24091992","10628820","1594323","0"],["0","-93312","-1026432","-4129056","-6718464","-1377810","6967782","5314410","1062882","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","-20736","-269568","-1679616","-6998400","-20356596","-36610380","-31591215","-354294","17006112","9565938","1594323"],["0","62208","777600","4105728","12667104","27687420","46491246","54147933","36315135","12223143","1594323","0"],["0","0","93312","1119744","5248800","11967264","13345074","6377292","1062882","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","373248","6718464","51648192","220449600","568287576","901323936","859340097","463947993","129140163","14348907"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","497664","9704448","81181440","377073792","1043671392","1676991600","1243571940","-366694290","-1477937421","-1114431777","-358722675","-43046721"],["0","0","0","0","746496","13436928","103296384","440899200","1136575152","1802647872","1718680194","927895986","258280326","28697814","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","-331776","-6469632","-52627968","-222269184","-448877376","73693152","2766799944","7259484060","9830595618","7754787072","3539397060","860934420","86093442"],["0","0","0","0","-995328","-20901888","-192595968","-1021206528","-3433974912","-7611179904","-11207027808","-10815887232","-6633978003","-2463229035","-502211745","-43046721"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","331776","7962624","85100544","532998144","2165864832","5971034880","11346855840","14812323552","13027213233","7455054348","2630632950","516560652","43046721"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","995328","20528640","184757760","949402944","3058580736","6381543528","8600841144","7252575327","3592009719","942244893","100442349"],["0","0","0","746496","13436928","103296384","440899200","1136575152","1802647872","1718680194","927895986","258280326","28697814"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","-663552","-14432256","-137728512","-755267328","-2614742208","-5897341728","-8580055896","-7552839492","-3196617615","299732724","908764110","344373768","43046721"],["0","0","0","-995328","-20901888","-192595968","-1021206528","-3433974912","-7611179904","-11207027808","-10815887232","-6633978003","-2463229035","-502211745","-43046721"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","331776","7962624","85100544","532998144","2165864832","5971034880","11346855840","14812323552","13027213233","7455054348","2630632950","516560652","43046721"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","5184","62208","291600","664848","741393","354294","59049"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-3"]]},"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 1) 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 >= 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":["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 [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + 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":"1","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":1,"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":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"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 >= 1"},"parent_proof":{"route":"liu_wang_three_term","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, 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":"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 >= 3, with A_n(x) = (2) + (3) x, V_n(x) = 0, W_n(x) = (1) + (3) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = 1 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 = 3: [x^0] W_n = 1 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 = 3: [x^0] W_n = 1 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 >= 3, with A_n(x) = (2) + (3) x, B_n(x) = (1) + (3) x: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, -1/3] (so its coefficients have one sign); for n >= 2, 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 >= 3","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":2,"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 >= 3, with A_n(x) = (2) + (3) x, B_n(x) = (1) + (3) x","every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, -1/3] (so its coefficients have one sign); for n >= 2, 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 >= 3"],"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 >= 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 = phi(n, D) = 1 + (1) D 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, 0]"},"kind":"preserver_image"}