{"id":"e-017120e86183432c","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9ac33c0e4aa39b7d","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"c2d5e2cc292636302e16b5cdf242ba303f3345d18df61e009bc18c9111b9f982","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14011,"certificate_compressed_size":2698,"conclusion_name":"basis_map_A007318 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","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-e073e5e0b021f36a","role":"sequence","name":"Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).","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]"}],"cited_by":[{"node_id":"n-9ac33c0e4aa39b7d","claim":"interlacing"},{"node_id":"n-9ac33c0e4aa39b7d","claim":"real_rooted"},{"node_id":"n-9ac33c0e4aa39b7d","claim":"one_signed_coefficients"},{"node_id":"n-9ac33c0e4aa39b7d","claim":"hadamard_kernel"}]}