{"input":{"first_index":0,"initial_polynomials":[["1"],["0","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","8"],["0","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"],["0","0","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"hadamard_factor","lift":{"factor":{"terms":[{"coefficient_offset":1,"index_offset":1,"kind":"binomial"}]},"factor_text":"binom(n+1, k+1)","gauge":{"denominator":["0","0","1"],"initial_value":"1","kind":"ratio","numerator":["1","2","1"]},"hidden_coefficients":"zero","identity":{"certificate":{"conjugated_input_relation":{"terms":[{"coeff":{"coeffs":[["0","0","-216"],["0","0","-744"],["0","0","-990"],["0","0","-636"],["0","0","-198"],["0","0","-24"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0","-216"],["0","0","0","-744"],["0","0","0","-990"],["0","0","0","-636"],["0","0","0","-198"],["0","0","0","-24"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","-36"],["0","0","0","0","-124"],["0","0","0","0","-165"],["0","0","0","0","-106"],["0","0","0","0","-33"],["0","0","0","0","-4"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0"],["0","256"],["0","640"],["0","640"],["0","320"],["0","80"],["0","8"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0"],["0","0","128"],["0","0","320"],["0","0","320"],["0","0","160"],["0","0","40"],["0","0","4"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["216"],["540"],["558"],["305"],["93"],["15"],["1"]]},"deriv_order":0,"shift":2}]},"conjugation_x_shift":0,"exceptional_bases":[],"factor":{"terms":[{"coefficient_offset":1,"index_offset":1,"kind":"binomial"}]},"gauge":{"denominator":["0","0","1"],"initial_value":"1","kind":"ratio","numerator":["1","2","1"]},"identifying_indices":[0,1],"input_leader_shift":2,"input_relation":{"terms":[{"coeff":{"coeffs":[["0","0","-9"],["0","0","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0"],["0","4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"input_valid_from_base":0,"multiplier":["1"],"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0","216"],["0","0","0","796"],["0","0","0","1198"],["0","0","0","945"],["0","0","0","413"],["0","0","0","95"],["0","0","0","9"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","128"],["0","0","0","0","320"],["0","0","0","0","320"],["0","0","0","0","160"],["0","0","0","0","40"],["0","0","0","0","4"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","0","216"],["0","0","540"],["0","0","558"],["0","0","305"],["0","0","93"],["0","0","15"],["0","0","1"]]},"deriv_order":0}],"shift":1}],"row_annihilator":[{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-1"]]},"deriv_order":1}],"row_annihilator_quotient":[{"coeff":{"coeffs":[["0","0","0","0","180"],["0","0","0","0","528"],["0","0","0","0","629"],["0","0","0","0","375"],["0","0","0","0","111"],["0","0","0","0","13"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","36"],["0","0","0","0","0","124"],["0","0","0","0","0","165"],["0","0","0","0","0","106"],["0","0","0","0","0","33"],["0","0","0","0","0","4"]]},"deriv_order":1}],"row_multiplier":["0","0","1"],"statement":"mu(x) w(n) M'' = sum_nu H_nu S^nu L + Z A (A the row annihilator of the product parent) with M'' the input relation conjugated by lambda(n) binom(n+1, k+1); so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 2 identifying rows gives Q_n = lambda(n) sum_k u(n,k) [x^k]P_n x^k for every n >= 0"},"status":"certified"},"lift_summary":{"hypotheses":[{"evidence":"binom(n+1, k+1): sum_k u(n,k) x^k / k! = L_(A-1)^(1)(-x) with A = n + a (zero when A < 1), a Laguerre polynomial with parameter 1 > -1, whose zeros are real and positive (Szego, Orthogonal Polynomials, Theorem 3.3.1); third condition of Theorem 40","name":"factor_multiplier_sequence","statement":"k -> binom(n+1, k+1) is a nonnegative multiplier sequence for every n >= 0","status":{"kind":"verified"}},{"evidence":"both ratio polynomials are nonzero at every integer n > n0 (exact sign checker) and lambda(n0) != 0","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"conjugation_division","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"the factor binom(n+1, k+1) depends on n, so consecutive rows are mapped by different operators","status":"not_proved"},"lean_replay":"not_attempted","operation":{"factor":"binom(n+1, k+1)","kind":"hadamard_factor"},"parent":{"gauge":{"denominator":["0","0","1"],"initial_value":"1","kind":"ratio","numerator":["1","2","1"]},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) sum_k binom(n+1, k+1) [x^k]P_n x^k 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]: a nonnegative multiplier sequence keeps real-rootedness and one coefficient sign","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":"Polya--Schur characterization of multiplier sequences (G. Polya and I. Schur, J. Reine Angew. Math. 144 (1914) 89--113; stated as Theorem 40 of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601): a real sequence (lambda_k) is a multiplier sequence (x^k -> lambda_k x^k maps real-rooted polynomials to real-rooted polynomials or zero) iff sum_k binom(m,k) lambda_k x^k is real-rooted with zeros of one sign for every m, iff sum_k lambda_k x^k / k! (or its reflection) is C x^m e^(a x) prod (1 + alpha_j x) with a, alpha_j >= 0; a product of multiplier sequences is one (composition), and a nonzero factor lambda(n) of the row changes nothing"},"verdict":"proved"},"multiplier_sequence":"binom(n+1, k+1): sum_k u(n,k) x^k / k! = L_(A-1)^(1)(-x) with A = n + a (zero when A < 1), a Laguerre polynomial with parameter 1 > -1, whose zeros are real and positive (Szego, Orthogonal Polynomials, Theorem 3.3.1); third condition of Theorem 40","parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"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}}