{"id":"n-9b40f045d129e237","revision":1,"first_index":0,"fingerprint":"f6-f83cf33d2c000fbc","name":"Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["3","2"],["7","8","2"],["18","30","15","2"],["47","104","80","24","2"],["123","340","355","170","35","2"],["322","1068","1410","932","315","48","2"],["843","3262","5208","4396","2079","532","63","2"],["2207","9760","18280","18784","11440","4144","840","80","2"],["5778","28746","61785","74838","55809","26226","7602","1260","99","2"],["15127","83620","202840","282980","249815","144488","54690","13080","1815","120","2"],["39603","240812","650705","1027466","1048135","720412","337183","105930","21384","2530","143","2"],["103682","687768","2048532","3611528","4180722","3330408","1858892","725208","193347","33528","3432","168","2"],["271443","1950650","6349200","12361960","16011450","14508520","9411350","4392700","1460030","336050","50765","4550","195","2"],["710647","5499704","19421150","41387920","59305960","60245080","44550660","24227400","9662835","2782780","560560","74620","5915","224","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","5","17","65","257","1025","4097","16385","65537","262145","1048577","4194305","16777217","67108865","268435457"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A122076","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159151,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-644899f192503d96"]}],"region_exact":{},"incoming_edges":[{"id":"e-644899f192503d96","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-9b40f045d129e237","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"8c15e5d9a2d8ca46f4b7159eeecf6f000deb6840bc3f6e5ffd71798d7d25ff18","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[]}],"used_by":[{"id":"e-58ea4b3a2d7c4054","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-04f443bf4d795a35","conclusion_name":"one_plus_derivative of Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows."},{"id":"e-895e752df1147f88","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-24403ee2e7d633a4","conclusion_name":"shift_x_plus_one of Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows."},{"id":"e-fa6dba774ff7b5f5","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-aa424fc3fffbd09d","conclusion_name":"derivative of Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows."}],"used_by_total":3,"links":[{"id":"l-651cdef839c89a7e","kind":"oeis_match","node_a_id":"n-9b40f045d129e237","node_b_id":null,"fingerprint":null,"oeis_id":"A123166","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":"A123166","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a8b114e0aff16b8c","kind":"oeis_match","node_a_id":"n-9b40f045d129e237","node_b_id":null,"fingerprint":null,"oeis_id":"A122076","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":53},"in_catalog":true,"oeis_id":"A122076","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b7d2186fe06758f3","kind":"oeis_match","node_a_id":"n-9b40f045d129e237","node_b_id":null,"fingerprint":null,"oeis_id":"A046231","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":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A046231","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e1f5c72ffaff3b66","kind":"oeis_match","node_a_id":"n-9b40f045d129e237","node_b_id":null,"fingerprint":null,"oeis_id":"A052539","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":"A052539","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f163a914ecdf634e","kind":"oeis_match","node_a_id":"n-9b40f045d129e237","node_b_id":null,"fingerprint":null,"oeis_id":"A092896","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":0,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A092896","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A122076","relation":"identification"},{"oeis_id":"A046231","relation":"match_row_sums"},{"oeis_id":"A052539","relation":"match_row_sums"},{"oeis_id":"A092896","relation":"match_row_sums"},{"oeis_id":"A123166","relation":"match_row_sums"},{"oeis_id":"A122076","relation":"match_triangle"}],"bfiles":[{"node_id":"n-9b40f045d129e237","oeis_id":"A052539","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":12972},{"node_id":"n-9b40f045d129e237","oeis_id":"A122076","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":53,"oeis_terms_agreeing":53,"oeis_terms_available":53,"term_count":990,"raw_size":17951}],"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:    2;\n  1:    3,     2;\n  2:    7,     8,     2;\n  3:   18,    30,    15,     2;\n  4:   47,   104,    80,    24,     2;\n  5:  123,   340,   355,   170,    35,     2;\n  6:  322,  1068,  1410,   932,   315,    48,    2;\n  7:  843,  3262,  5208,  4396,  2079,   532,   63,    2;\n  8: 2207,  9760, 18280, 18784, 11440,  4144,  840,   80,  2;\n  9: 5778, 28746, 61785, 74838, 55809, 26226, 7602, 1260, 99, 2;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] - (1 + 2*x + x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] - (1 + 2*x + x^2)*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}},{"oeis_id":"A122076","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    2;\n  1:    3,     2;\n  2:    7,     8,     2;\n  3:   18,    30,    15,     2;\n  4:   47,   104,    80,    24,     2;\n  5:  123,   340,   355,   170,    35,     2;\n  6:  322,  1068,  1410,   932,   315,    48,    2;\n  7:  843,  3262,  5208,  4396,  2079,   532,   63,    2;\n  8: 2207,  9760, 18280, 18784, 11440,  4144,  840,   80,  2;\n  9: 5778, 28746, 61785, 74838, 55809, 26226, 7602, 1260, 99, 2;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] - (1 + 2*x + x^2)*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] - (1 + 2*x + x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-9b40f045d129e237","steps":[],"edges":[],"text":"A122076"},"headline":"A122076: Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","chain":null}
