{"id":"e-61a30fbd1c55d0f9","operation":{"factor":"1/(n-k)!","operation":"hadamard_factor"},"operation_detail":"1/(n-k)!","conclusion_id":"n-511a2225fa2191c9","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":"abaed2f314831bb42f9170038bf909154a7179edd57da15cd7c6c58c8f359833","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"hadamard_factor","certificate_raw_size":5998,"certificate_compressed_size":1740,"conclusion_name":"Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","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]"}],"cited_by":[{"node_id":"n-511a2225fa2191c9","claim":"real_rooted"},{"node_id":"n-511a2225fa2191c9","claim":"one_signed_coefficients"},{"node_id":"n-511a2225fa2191c9","claim":"hadamard_kernel"}]}