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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","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"deb72030a64566f62c9ea0b7edfa0b9f2606f5bc228f87c94169bc8c82e4be8c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9ed87ca196739b1c","role":"sequence","name":"Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1f9675a83f1331a4","kind":"fingerprint_related","node_a_id":"n-686dc44c06bd803d","node_b_id":"n-88fe5d645a597110","fingerprint":"f6-afe77babea4b595f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-afe77babea4b595f"},"other_id":"n-88fe5d645a597110","other":{"id":"n-88fe5d645a597110","name":"theta_plus_one_squared of Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. 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