{"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","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","2","-1"],["2","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["1","0","-2","0","1"],["2","-2","-2","2","0"],["1","-2","1","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","4","1","-2"],["-2","4","-2","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1","-2","1"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["-1","-2","-1","0","0"],["2","6","6","2","0"],["-1","-4","-6","-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","-1","1","1"],["2","6","6","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["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","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","2","-1"],["2","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["1","-2","1"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","2","1"],["2","2","0"],["1","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","-2"],["-2","0"]]},"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 2 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","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","2","-1"],["2","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","2"],["2","0"]]},"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":{"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 [-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":"open"},"notes":[],"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: coefficients must be index-independent; 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) is not of the form gamma(n) w(x)","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 n - 1) + (2) x, V_n(x) = 0, W_n(x) = (-n^2 + 2 n - 1) + (-2 n + 2) x + (-1) x^2; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = -n^2 + 2 n - 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 = 1 (not a disproof): fails at n = 2: [x^1] 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 = 1 (not a disproof): fails at n = 2: [x^1] 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 n - 1) + (2) x, B_n(x) = (-n^2 + 2 n - 1) + (-2 n + 2) x + (-1) x^2: every P_n, n >= 0, is nonzero and real-rooted; 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; the theorem certifies no root region, so the roots are not contained in (-inf, 0], so the rows are real-rooted but this route does not give coefficients of one sign","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"no lean_proved catalog entry equals the parent; 25 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A029653:lean_proved, oeis_catalog:A060920:lean_proved, oeis_catalog:A084950:lean_proved, oeis_catalog:A093375:lean_proved, oeis_catalog:A104029:lean_proved, oeis_catalog:A106246:lean_proved, oeis_catalog:A113214:lean_proved, oeis_catalog:A113413:lean_proved, oeis_catalog:A117317:lean_proved, oeis_catalog:A119308:lean_proved, oeis_catalog:A119808:lean_proved, oeis_catalog:A121574:lean_proved, oeis_catalog:A126216:lean_proved, oeis_catalog:A133087:lean_proved, oeis_catalog:A134082:lean_proved, oeis_catalog:A154929:lean_proved, oeis_catalog:A156319:lean_proved, oeis_catalog:A167194:lean_proved, oeis_catalog:A193727:lean_proved, oeis_catalog:A193728:lean_proved, oeis_catalog:A193730:lean_proved, oeis_catalog:A193735:lean_proved, oeis_catalog:A202241:lean_proved, oeis_catalog:A318397:lean_proved, oeis_catalog:A394813: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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 0) 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"}