{"id":"n-a5bdea4e5eb1caa8","revision":1,"first_index":0,"fingerprint":"f6-19c649f255ec1907","name":"shift_x_plus_one of Triangle read by rows, A007318 rows repeated three times.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1"],["1"],["2","1"],["2","1"],["2","1"],["4","4","1"],["4","4","1"],["4","4","1"],["8","12","6","1"],["8","12","6","1"],["8","12","6","1"],["16","32","24","8","1"],["16","32","24","8","1"],["16","32","24","8","1"]],"degrees":[0,0,0,1,1,1,2,2,2,3,3,3,4,4,4],"row_sums":["1","1","1","3","3","3","9","9","9","27","27","27","81","81","81"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160711,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-345c88cc98fbdbfc"]},{"id":160712,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-131d94ab8eb56aaf","e-345c88cc98fbdbfc"]},{"id":160713,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-131d94ab8eb56aaf","e-345c88cc98fbdbfc"]},{"id":160714,"position":3,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-131d94ab8eb56aaf"]}],"region_exact":{},"incoming_edges":[{"id":"e-131d94ab8eb56aaf","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-a5bdea4e5eb1caa8","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":"fb583593f6094da1355fb5099555b13fe51c821f837f2f4935ffa7ea9b6226d0","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-99e5bf113a0a46cd","role":"sequence","name":"Triangle read by rows, A007318 rows repeated three times.","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]"}]},{"id":"e-345c88cc98fbdbfc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-a5bdea4e5eb1caa8","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"01db4a5399c1098c3f1974a102a9174d5353ae64a22ff661f5ebc5890bf7846f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-99e5bf113a0a46cd","role":"sequence","name":"Triangle read by rows, A007318 rows repeated three times.","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-025eb774f7b3606a","kind":"fingerprint_related","node_a_id":"n-a5bdea4e5eb1caa8","node_b_id":"n-ae337235b402d9c7","fingerprint":"f6-19c649f255ec1907","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-19c649f255ec1907"},"other_id":"n-ae337235b402d9c7","other":{"id":"n-ae337235b402d9c7","name":"basis_map_A013611 of Triangle read by rows, A007318 rows repeated three times.","first_index":0,"fingerprint":"f6-19c649f255ec1907","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1"]],"card_rows":[["1"],["1"],["1"],["2","4"]],"row_sums_preview":["1","1","1","6","6","6","36","36"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-99e5bf113a0a46cd","steps":["basis A013611"],"edges":["e-15d22f53a2fb1793"],"text":"A152828 \u2192 basis A013611"},"headline":"A152828 \u2192 basis A013611","chain":null}},{"id":"l-84c1d257a3d19c83","kind":"oeis_match","node_a_id":"n-a5bdea4e5eb1caa8","node_b_id":null,"fingerprint":null,"oeis_id":"A127975","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A127975","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A127975","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-a5bdea4e5eb1caa8","oeis_id":"A127975","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":200,"raw_size":4125}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1;\n  3: 2,  1;\n  4: 2,  1;\n  5: 2,  1;\n  6: 4,  4, 1;\n  7: 4,  4, 1;\n  8: 4,  4, 1;\n  9: 8, 12, 6, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 3, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (2 + x)*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-99e5bf113a0a46cd","steps":["basis A007318"],"edges":["e-131d94ab8eb56aaf"],"text":"A152828 \u2192 basis A007318"},"headline":"A152828 \u2192 basis A007318","chain":null}
