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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(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":"Q_0 = 1, Q_n = Q_(n-1) + w(n) t Q_(max(0, n-2)), w(n) = -1 + n (paper_proof_not_lean_checked): For every n >= 1: Q_n has nonnegative coefficients and constant term 1, is real-rooted, and Q_i << Q_n (oriented weak interlacing) for p(n) <= i <= n. Hence Q_A << Q_B whenever 0 <= A <= B and B - A <= 2.; the identifying seeds equal the canonical rows (Canonical), so every parent row is a canonical row","route":"paper_theorem","success":true,"variant":"fixed_delay_predecessor_window"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["p(n) = max(0, n-2) with 2 >= 1 is nondecreasing and 0 <= p(n) < n for n >= 1 (exact_structural)","w(n) >= 0 for every integer n >= 1: w(1+m) has nonnegative coefficients [0, 1] in m (exact_symbolic_certificate)","window: Q_A << Q_B for 0 <= A <= B with B - A <= 2 (B - A <= 2)","the identifying parent seeds equal the canonical rows (Canonical), so every parent row is a canonical row and consecutive parent rows are consecutive canonical rows"],"theorem":"fixed_delay_predecessor_window (paper_proof_not_lean_checked; https://github.com/PerAlexandersson/RealRooted/issues/2037): For every n >= 1: Q_n has nonnegative coefficients and constant term 1, is real-rooted, and Q_i << Q_n (oriented weak interlacing) for p(n) <= i <= n. 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