{"id":"e-aaeb3397827b8bbd","operation":{"index_shift":2,"operation":"preserver_image","operator":"{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"binomial\",\"index_offset\":2,\"coefficient_offset\":1}]}}"},"operation_detail":"hadamard_multiplier, shift 2","conclusion_id":"n-05b91207a1ba3acf","premise_count":1,"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","lean_route":null,"lean_declarations":[],"certificate_hash":"ae7baffb4fb7f45b3e62a1927fda890eb1941f01579fafb7f0af89ec9600f404","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":28056,"certificate_compressed_size":6506,"conclusion_name":"Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c963abfc0266d66c","role":"sequence","name":"Triangle of Dyck paths counted by number of long interior inclines.","uses":{"claim":"real_rooted","from_index":3,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-05b91207a1ba3acf","claim":"real_rooted"},{"node_id":"n-05b91207a1ba3acf","claim":"one_signed_coefficients"},{"node_id":"n-05b91207a1ba3acf","claim":"hadamard_kernel"}]}