{"id":"n-d7abbe24afd98bff","revision":0,"first_index":0,"fingerprint":"f6-37c3559a5d3b5b73","name":"Riordan array (1, x/(1-3*x+2*x^2)).","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":[["1"],["0","1"],["0","3","1"],["0","7","6","1"],["0","15","23","9","1"],["0","31","72","48","12","1"],["0","63","201","198","82","15","1"],["0","127","522","699","420","125","18","1"],["0","255","1291","2223","1795","765","177","21","1"],["0","511","3084","6562","6768","3840","1260","238","24","1"],["0","1023","7181","18324","23276","16758","7266","1932","308","27","1"],["0","2047","16398","49029","74616","65870","36036","12586","2808","387","30","1"],["0","4095","36879","126837","226325","238710","159446","69930","20394","3915","475","33","1"],["0","8191","81936","319332","656580","810715","644976","344064","125496","31365","5280","572","36","1"],["0","16383","180241","786258","1836422","2611305","2426751","1537308","679764","211761","46255","6930","678","39","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","4","14","48","164","560","1912","6528","22288","76096","259808","887040","3028544","10340096"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A206306","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"],["0","3","1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-6","-2"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4"],["-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169711,"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-2d1e5381bf0c7c76"]},{"id":169712,"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-2d1e5381bf0c7c76"]},{"id":169713,"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-2d1e5381bf0c7c76"]},{"id":169714,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"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 >= 2","region_exact":null,"edges":["e-2d1e5381bf0c7c76"]}],"region_exact":{},"incoming_edges":[{"id":"e-2d1e5381bf0c7c76","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-d7abbe24afd98bff","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_eval_nonpos","RealRooted.threeTerm_interlaces","RealRooted.eval_nonpos_of_natDegree_le_two"],"certificate_hash":"a21a01acb424176057fc7cf69b4125fefe4bbb2e52470573b659fdda6fd66094","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-03a712bd8e28a7a7","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-39e3865552b07edc","conclusion_name":"basis_map_A008277 of Riordan array (1, x/(1-3*x+2*x^2))."},{"id":"e-3e6bcb0c5ceb0c12","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-f6a21c7b563b30ff","conclusion_name":"basis_map_A007318 of Riordan array (1, x/(1-3*x+2*x^2))."},{"id":"e-6a6a60f8cc707cd2","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-cc618c8c4d7fd590","conclusion_name":"basis_map_A013611 of Riordan array (1, x/(1-3*x+2*x^2))."},{"id":"e-bce77ed0b34406cb","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-74ec83953eae1292","conclusion_name":"basis_map_A013610 of Riordan array (1, x/(1-3*x+2*x^2))."},{"id":"e-e8ca8860a1636b06","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-a8289b7270086fb2","conclusion_name":"basis_map_A013609 of Riordan array (1, x/(1-3*x+2*x^2))."},{"id":"e-44916467ec069569","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-6f13d958f71d4ca4","conclusion_name":"derivative of Riordan array (1, x/(1-3*x+2*x^2))."}],"used_by_total":6,"links":[{"id":"l-8a67ddb5a090f411","kind":"fingerprint_related","node_a_id":"n-9b5c7e97e5191da6","node_b_id":"n-d7abbe24afd98bff","fingerprint":"f6-37c3559a5d3b5b73","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-37c3559a5d3b5b73"},"other_id":"n-9b5c7e97e5191da6","other":{"id":"n-9b5c7e97e5191da6","name":"Triangular array formed by the Mersenne numbers.","first_index":0,"fingerprint":"f6-37c3559a5d3b5b73","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["7","6","1"]],"card_rows":[["1"],["3","1"],["7","6","1"],["15","23","9","1"]],"row_sums_preview":["1","4","14","48","164","560","1912","6528"],"regions":["(-inf, +inf)","(-inf, 0)","(-6, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A110441"],"derivation":{"root":"n-9b5c7e97e5191da6","steps":[],"edges":[],"text":"A110441"},"headline":"A110441: Triangular array formed by the Mersenne numbers.","chain":null}},{"id":"l-9582905661502e9b","kind":"fingerprint_related","node_a_id":"n-369f42b9099fbd84","node_b_id":"n-d7abbe24afd98bff","fingerprint":"f6-37c3559a5d3b5b73","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-37c3559a5d3b5b73"},"other_id":"n-369f42b9099fbd84","other":{"id":"n-369f42b9099fbd84","name":"reversal of Triangular array formed by the Mersenne numbers.","first_index":0,"fingerprint":"f6-37c3559a5d3b5b73","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","6","7"]],"card_rows":[["1"],["1","3"],["1","6","7"],["1","9","23","15"]],"row_sums_preview":["1","4","14","48","164","560","1912","6528"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9b5c7e97e5191da6","steps":["reverse"],"edges":["e-5faf1c18373ba45e"],"text":"A110441 \u2192 reverse"},"headline":"A110441 \u2192 reverse","chain":null}},{"id":"l-7c4e1b854e89285d","kind":"oeis_match","node_a_id":"n-d7abbe24afd98bff","node_b_id":null,"fingerprint":null,"oeis_id":"A204089","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":"A204089","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-af7be1b44b4da4c9","kind":"oeis_match","node_a_id":"n-d7abbe24afd98bff","node_b_id":null,"fingerprint":null,"oeis_id":"A206306","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A206306","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ca368ebd01e29bb0","kind":"oeis_match","node_a_id":"n-d7abbe24afd98bff","node_b_id":null,"fingerprint":null,"oeis_id":"A007070","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":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A007070","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-da8b711ec9dfe06d","kind":"oeis_match","node_a_id":"n-d7abbe24afd98bff","node_b_id":null,"fingerprint":null,"oeis_id":"A110441","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A110441","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A206306","relation":"identification"},{"oeis_id":"A007070","relation":"match_row_sums"},{"oeis_id":"A204089","relation":"match_row_sums"},{"oeis_id":"A110441","relation":"match_triangle"},{"oeis_id":"A206306","relation":"match_triangle"}],"bfiles":[{"node_id":"n-d7abbe24afd98bff","oeis_id":"A007070","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":28,"oeis_terms_available":28,"term_count":200,"raw_size":11624},{"node_id":"n-d7abbe24afd98bff","oeis_id":"A110441","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16076},{"node_id":"n-d7abbe24afd98bff","oeis_id":"A204089","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":28,"oeis_terms_available":28,"term_count":200,"raw_size":11518},{"node_id":"n-d7abbe24afd98bff","oeis_id":"A206306","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":15369}],"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: 0,   1;\n  2: 0,   3,    1;\n  3: 0,   7,    6,    1;\n  4: 0,  15,   23,    9,    1;\n  5: 0,  31,   72,   48,   12,    1;\n  6: 0,  63,  201,  198,   82,   15,    1;\n  7: 0, 127,  522,  699,  420,  125,   18,   1;\n  8: 0, 255, 1291, 2223, 1795,  765,  177,  21,  1;\n  9: 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(n-2)*x)*P(n-1,x) - (2*n-4)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 3*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[2, x_] := 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (n - 2)*x)*P[n - 1, x] - (2*n - 4)*P[n - 2, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; v[3] = 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 6 + (n - 2)*x)*v[i - 1] - (2*n - 4)*v[i - 2])/(n - 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":"A110441","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    3,    1;\n  2:    7,    6,     1;\n  3:   15,   23,     9,     1;\n  4:   31,   72,    48,    12,     1;\n  5:   63,  201,   198,    82,    15,    1;\n  6:  127,  522,   699,   420,   125,   18,    1;\n  7:  255, 1291,  2223,  1795,   765,  177,   21,   1;\n  8:  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;\n  9: 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (n-1)*P(n,x) = (3*n-3+(n-1)*x)*P(n-1,x) - (2*n-2)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 3*x + x^2.","mathematica":"P[0, x_] := x;\nP[1, x_] := 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (n - 1)*x)*P[n - 1, x] - (2*n - 2)*P[n - 2, x])/(n - 1)]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (n - 1)*x)*v[i - 1] - (2*n - 2)*v[i - 2])/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":1,"offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A206306","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: 1;\n  1: 0,   1;\n  2: 0,   3,    1;\n  3: 0,   7,    6,    1;\n  4: 0,  15,   23,    9,    1;\n  5: 0,  31,   72,   48,   12,    1;\n  6: 0,  63,  201,  198,   82,   15,    1;\n  7: 0, 127,  522,  699,  420,  125,   18,   1;\n  8: 0, 255, 1291, 2223, 1795,  765,  177,  21,  1;\n  9: 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(n-2)*x)*P(n-1,x) - (2*n-4)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 3*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[2, x_] := 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (n - 2)*x)*P[n - 1, x] - (2*n - 4)*P[n - 2, x])/(n - 2)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; v[3] = 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 6 + (n - 2)*x)*v[i - 1] - (2*n - 4)*v[i - 2])/(n - 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-d7abbe24afd98bff","steps":[],"edges":[],"text":"A206306"},"headline":"A206306: Riordan array (1, x/(1-3*x+2*x^2)).","chain":null}
