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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":"517634bd991548fae3b9764476718c2a0a3ad0e09618f2fc72b51d650f569fe0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3af50523900b3357","role":"sequence","name":"basis_map_A013611 of narayana","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-bca6b9c2ada49986","kind":"fingerprint_related","node_a_id":"n-b991dcbedf3c1d83","node_b_id":"n-035bf07f85d18538","fingerprint":"f6-777088b2ee260e77","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-777088b2ee260e77"},"other_id":"n-b991dcbedf3c1d83","other":{"id":"n-b991dcbedf3c1d83","name":"theta_plus_one_squared of basis_map_A013611 of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. 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