{"id":"n-f239c3b0ba932394","revision":0,"first_index":0,"fingerprint":"f6-1cfd4c48d5eaa881","name":"Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","1"],["0","1","1"],["0","1","2","1"],["0","0","2","3","1"],["0","0","1","4","4","1"],["0","0","0","3","7","5","1"],["0","0","0","1","7","11","6","1"],["0","0","0","0","4","14","16","7","1"],["0","0","0","0","1","11","25","22","8","1"],["0","0","0","0","0","5","25","41","29","9","1"],["0","0","0","0","0","1","16","50","63","37","10","1"],["0","0","0","0","0","0","6","41","91","92","46","11","1"],["0","0","0","0","0","0","1","22","91","154","129","56","12","1"],["0","0","0","0","0","0","0","7","63","182","246","175","67","13","1"],["0","0","0","0","0","0","0","1","29","154","336","375","231","79","14","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["2","2","4","6","10","16","26","42","68","110","178","288","466","754","1220"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A199881","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","1"],["0","1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169681,"position":0,"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-8ff02e4505879db1"]},{"id":169682,"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-8ff02e4505879db1"]},{"id":169683,"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-8ff02e4505879db1"]},{"id":169684,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-8ff02e4505879db1"]}],"region_exact":{},"incoming_edges":[{"id":"e-8ff02e4505879db1","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-f239c3b0ba932394","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.threeTerm_interlaces","RealRooted.threeTerm_hasNonnegCoeffs"],"certificate_hash":"4a4b4fb2c5cb31be31ce08ba3712b00c81f312eaba961ccef2765c6f1113f9d7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-12cac8e5e3e8e9fc","kind":"oeis_match","node_a_id":"n-f239c3b0ba932394","node_b_id":null,"fingerprint":null,"oeis_id":"A128588","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A128588","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-19789ff774810fd6","kind":"oeis_match","node_a_id":"n-f239c3b0ba932394","node_b_id":null,"fingerprint":null,"oeis_id":"A163733","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":1,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A163733","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-37d53054273704d9","kind":"oeis_match","node_a_id":"n-f239c3b0ba932394","node_b_id":null,"fingerprint":null,"oeis_id":"A055389","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A055389","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-67fd4875b196ae42","kind":"oeis_match","node_a_id":"n-f239c3b0ba932394","node_b_id":null,"fingerprint":null,"oeis_id":"A078642","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A078642","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8adec6b7d100f398","kind":"oeis_match","node_a_id":"n-f239c3b0ba932394","node_b_id":null,"fingerprint":null,"oeis_id":"A198834","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":1,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A198834","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d1f8eb52d6bdb06b","kind":"oeis_match","node_a_id":"n-f239c3b0ba932394","node_b_id":null,"fingerprint":null,"oeis_id":"A006355","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A006355","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f102ac26afcaea29","kind":"oeis_match","node_a_id":"n-f239c3b0ba932394","node_b_id":null,"fingerprint":null,"oeis_id":"A068922","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A068922","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f7986fe51c4376cb","kind":"oeis_match","node_a_id":"n-f239c3b0ba932394","node_b_id":null,"fingerprint":null,"oeis_id":"A118658","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A118658","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A006355","relation":"match_row_sums"},{"oeis_id":"A055389","relation":"match_row_sums"},{"oeis_id":"A068922","relation":"match_row_sums"},{"oeis_id":"A078642","relation":"match_row_sums"},{"oeis_id":"A118658","relation":"match_row_sums"},{"oeis_id":"A128588","relation":"match_row_sums"},{"oeis_id":"A163733","relation":"match_row_sums"},{"oeis_id":"A198834","relation":"match_row_sums"},{"oeis_id":"A199881","relation":"tail"}],"bfiles":[{"node_id":"n-f239c3b0ba932394","oeis_id":"A078642","target":"row_sums","oeis_offset":1,"first_row":2,"rows_used":200,"terms_matched":13,"oeis_terms_agreeing":37,"oeis_terms_available":37,"term_count":200,"raw_size":5267}],"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, 1;\n  1: 0, 1, 1;\n  2: 0, 1, 2, 1;\n  3: 0, 0, 2, 3, 1;\n  4: 0, 0, 1, 4, 4,  1;\n  5: 0, 0, 0, 3, 7,  5,  1;\n  6: 0, 0, 0, 1, 7, 11,  6,  1;\n  7: 0, 0, 0, 0, 4, 14, 16,  7,  1;\n  8: 0, 0, 0, 0, 1, 11, 25, 22,  8, 1;\n  9: 0, 0, 0, 0, 0,  5, 25, 41, 29, 9, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1 + x, P(1,x) = x + x^2.","mathematica":"P[0, x_] := 1 + x;\nP[1, x_] := x + x^2;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; v[2] = x + x^2; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + x*v[i - 2]); 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-f239c3b0ba932394","steps":[],"edges":[],"text":"A199881"},"headline":"Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","chain":null}