{"id":"n-e7c079f81f452bf3","revision":0,"first_index":0,"fingerprint":"f6-c12a5875baa63cc4","name":"Another version of Fibonacci-Pascal triangle A037027.","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","1","2"],["0","0","2","3"],["0","0","1","5","5"],["0","0","0","3","10","8"],["0","0","0","1","9","20","13"],["0","0","0","0","4","22","38","21"],["0","0","0","0","1","14","51","71","34"],["0","0","0","0","0","5","40","111","130","55"],["0","0","0","0","0","1","20","105","233","235","89"],["0","0","0","0","0","0","6","65","256","474","420","144"],["0","0","0","0","0","0","1","27","190","594","942","744","233"],["0","0","0","0","0","0","0","7","98","511","1324","1836","1308","377"],["0","0","0","0","0","0","0","1","35","315","1295","2860","3522","2285","610"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","3","5","11","21","43","85","171","341","683","1365","2731","5461","10923"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A111006","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169334,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-a27f4f813bd9c6e4"]},{"id":169335,"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-a27f4f813bd9c6e4"]},{"id":169336,"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-a27f4f813bd9c6e4"]},{"id":169337,"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-a27f4f813bd9c6e4"]}],"region_exact":{},"incoming_edges":[{"id":"e-a27f4f813bd9c6e4","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-e7c079f81f452bf3","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":"window","lean_declarations":["RealRooted.threeTerm_interlaces_of_roots_mem_Icc"],"certificate_hash":"eb025caabae375ccf6e37363c67aef3e48f4da1a73beddf0ad9fbe4d3e677d8e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-0767c9a9bed43370","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-233801744e594f7f","conclusion_name":"shift_x_plus_one of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-0c8a8985960dccd5","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-3d2e989221bae0ab","conclusion_name":"basis_map_A008277 of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-7fed9254e8fe90c1","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-908d9fd8f65f4e8f","conclusion_name":"basis_map_A013609 of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-a9970c74218754b4","operation":"basis_map","operation_detail":"A008288","conclusion_id":"n-ff2c124f57c19690","conclusion_name":"basis_map_A008288 of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-b0fcc165eb472062","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-bb552e112dcbf0cb","conclusion_name":"basis_map_A013611 of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-fe606963c46c5c72","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-53f3ce2888a3b03b","conclusion_name":"basis_map_A013610 of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-041079b53a5d7e1b","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-26ecae263b1ee6f4","conclusion_name":"one_plus_derivative of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-46c0275dfe2e49c4","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-3ac4a93aea3ff306","conclusion_name":"theta_plus_one of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-48c384a60492b25b","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-ce9f6d305461a3cc","conclusion_name":"theta_plus_one_squared of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-537bf9237ec77614","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-a25d5a5440f7c216","conclusion_name":"mobius_x_over_x_plus_one of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-6598ad7955620559","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-f84265f03f8449db","conclusion_name":"derivative of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-80584d04caa9a628","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-233801744e594f7f","conclusion_name":"shift_x_plus_one of Another version of Fibonacci-Pascal triangle A037027."},{"id":"e-9f4ef744e9f7773b","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-7d332fcb7bf9a0f7","conclusion_name":"Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2))."}],"used_by_total":13,"links":[{"id":"l-379ee029a13c281f","kind":"fingerprint_related","node_a_id":"n-7d332fcb7bf9a0f7","node_b_id":"n-e7c079f81f452bf3","fingerprint":"f6-c12a5875baa63cc4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c12a5875baa63cc4"},"other_id":"n-7d332fcb7bf9a0f7","other":{"id":"n-7d332fcb7bf9a0f7","name":"Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-c12a5875baa63cc4","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["3","2"]],"row_sums_preview":["1","1","3","5","11","21","43","85"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":["A128100"],"derivation":{"root":"n-7d332fcb7bf9a0f7","steps":[],"edges":[],"text":"A128100"},"headline":"A128100: Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).","chain":null}},{"id":"l-6a335c200639c6d0","kind":"fingerprint_related","node_a_id":"n-dbc168630d629d01","node_b_id":"n-e7c079f81f452bf3","fingerprint":"f6-c12a5875baa63cc4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c12a5875baa63cc4"},"other_id":"n-dbc168630d629d01","other":{"id":"n-dbc168630d629d01","name":"mobius_x_over_x_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-c12a5875baa63cc4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","0","2","3"]],"row_sums_preview":["1","1","3","5","11","21","43","85"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-392f49b496f79fbf","steps":["x \u21a6 x/(x+1)"],"edges":["e-3f0133638547078b"],"text":"A026729 \u2192 x \u21a6 x/(x+1)"},"headline":"A026729 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-6b12ad61f38f3069","kind":"fingerprint_related","node_a_id":"n-e7c079f81f452bf3","node_b_id":"n-a1cb8cca853be542","fingerprint":"f6-c12a5875baa63cc4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c12a5875baa63cc4"},"other_id":"n-a1cb8cca853be542","other":{"id":"n-a1cb8cca853be542","name":"mobius_x_over_x_plus_one of theta_plus_one of Riordan array (1+2x,x(1+x)).","first_index":0,"fingerprint":"f6-c12a5875baa63cc4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["0","6","9"]],"card_rows":[["1"],["2","4"],["0","6","9"],["0","4","20","20"]],"row_sums_preview":["1","6","15","44","105","258","595","1368"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c27b11147fce8bfa","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-0a8a62dc4e77e317","e-972fbf1509bd1c6b"],"text":"A113214 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A113214 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7bdd4f9af51f519e","kind":"fingerprint_related","node_a_id":"n-359dae48f9937424","node_b_id":"n-e7c079f81f452bf3","fingerprint":"f6-c12a5875baa63cc4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c12a5875baa63cc4"},"other_id":"n-359dae48f9937424","other":{"id":"n-359dae48f9937424","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).","first_index":0,"fingerprint":"f6-c12a5875baa63cc4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["0","2","3"]],"card_rows":[["1"],["1","2"],["0","2","3"],["0","1","5","5"]],"row_sums_preview":["1","3","5","11","21","43","85","171"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-21854f4257effb01","steps":["x \u21a6 x/(x+1)"],"edges":["e-5f893429686fee24"],"text":"A030528 \u2192 x \u21a6 x/(x+1)"},"headline":"A030528 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-851e1ce773408316","kind":"fingerprint_related","node_a_id":"n-aeb67fdaad2bc42e","node_b_id":"n-e7c079f81f452bf3","fingerprint":"f6-c12a5875baa63cc4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c12a5875baa63cc4"},"other_id":"n-aeb67fdaad2bc42e","other":{"id":"n-aeb67fdaad2bc42e","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","first_index":0,"fingerprint":"f6-c12a5875baa63cc4","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["3","2"]],"card_rows":[["1"],["2","1"],["3","2"],["5","5","1"]],"row_sums_preview":["1","3","5","11","21","43","85","171"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-569a801b809b6abd","steps":["x \u21a6 x+1"],"edges":["e-124c90dc6fe1d800"],"text":"A169803 \u2192 x \u21a6 x+1"},"headline":"A169803 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-dc28e895e341cbe0","kind":"fingerprint_related","node_a_id":"n-622af52a4165c278","node_b_id":"n-e7c079f81f452bf3","fingerprint":"f6-c12a5875baa63cc4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c12a5875baa63cc4"},"other_id":"n-622af52a4165c278","other":{"id":"n-622af52a4165c278","name":"shift_x_plus_one of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","first_index":1,"fingerprint":"f6-c12a5875baa63cc4","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["3","2"]],"row_sums_preview":["1","1","3","5","11","21","43","85"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8b3f39d03de798f","steps":["x \u21a6 x+1"],"edges":["e-13fa50edf65b735e"],"text":"A052553 \u2192 x \u21a6 x+1"},"headline":"A052553 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-e2fbe02a028c3c22","kind":"fingerprint_related","node_a_id":"n-e7c079f81f452bf3","node_b_id":"n-ef1acb736e5c4a3b","fingerprint":"f6-c12a5875baa63cc4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c12a5875baa63cc4"},"other_id":"n-ef1acb736e5c4a3b","other":{"id":"n-ef1acb736e5c4a3b","name":"derivative of Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k).","first_index":0,"fingerprint":"f6-c12a5875baa63cc4","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["3"],["8","4"],["15","10"]],"row_sums_preview":["0","0","2","3","12","25","66","147"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-299efe8218728fe3","steps":["D"],"edges":["e-a4db8fbaa61eae59"],"text":"A245962 \u2192 D"},"headline":"A245962 \u2192 D","chain":null}},{"id":"l-04c6933ebafa27f8","kind":"oeis_match","node_a_id":"n-e7c079f81f452bf3","node_b_id":null,"fingerprint":null,"oeis_id":"A152046","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":"A152046","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-3665885374a98768","kind":"oeis_match","node_a_id":"n-e7c079f81f452bf3","node_b_id":null,"fingerprint":null,"oeis_id":"A001045","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":"A001045","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-63f718e4461a6115","kind":"oeis_match","node_a_id":"n-e7c079f81f452bf3","node_b_id":null,"fingerprint":null,"oeis_id":"A168642","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A168642","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6c89c02426f1d5d1","kind":"oeis_match","node_a_id":"n-e7c079f81f452bf3","node_b_id":null,"fingerprint":null,"oeis_id":"A167167","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"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":"A167167","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-9ffd567ea6c50bdd","kind":"oeis_match","node_a_id":"n-e7c079f81f452bf3","node_b_id":null,"fingerprint":null,"oeis_id":"A128100","in_catalog":1,"target":"reversed_triangle","row_alignment":"node_lengths","rows_matched":15,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":15,"target":"reversed_triangle","terms_compared":64},"in_catalog":true,"oeis_id":"A128100","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c9daf6cd1d15064d","kind":"oeis_match","node_a_id":"n-e7c079f81f452bf3","node_b_id":null,"fingerprint":null,"oeis_id":"A111006","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":12,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":12,"target":"triangle","terms_compared":78},"in_catalog":true,"oeis_id":"A111006","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e7be7e28d0ef84e0","kind":"oeis_match","node_a_id":"n-e7c079f81f452bf3","node_b_id":null,"fingerprint":null,"oeis_id":"A283642","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":"A283642","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A111006","relation":"identification"},{"oeis_id":"A128100","relation":"match_reversed_triangle"},{"oeis_id":"A001045","relation":"match_row_sums"},{"oeis_id":"A152046","relation":"match_row_sums"},{"oeis_id":"A167167","relation":"match_row_sums"},{"oeis_id":"A168642","relation":"match_row_sums"},{"oeis_id":"A283642","relation":"match_row_sums"},{"oeis_id":"A111006","relation":"match_triangle"}],"bfiles":[{"node_id":"n-e7c079f81f452bf3","oeis_id":"A111006","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":78,"oeis_terms_agreeing":78,"oeis_terms_available":78,"term_count":990,"raw_size":8600},{"node_id":"n-e7c079f81f452bf3","oeis_id":"A128100","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":62,"terms_matched":64,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":992,"raw_size":14013},{"node_id":"n-e7c079f81f452bf3","oeis_id":"A283642","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":34,"oeis_terms_available":34,"term_count":200,"raw_size":7006}],"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, 1, 2;\n  3: 0, 0, 2, 3;\n  4: 0, 0, 1, 5,  5;\n  5: 0, 0, 0, 3, 10,  8;\n  6: 0, 0, 0, 1,  9, 20, 13;\n  7: 0, 0, 0, 0,  4, 22, 38,  21;\n  8: 0, 0, 0, 0,  1, 14, 51,  71,  34;\n  9: 0, 0, 0, 0,  0,  5, 40, 111, 130, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + (x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (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] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (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":"A111006","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, 1, 2;\n  3: 0, 0, 2, 3;\n  4: 0, 0, 1, 5,  5;\n  5: 0, 0, 0, 3, 10,  8;\n  6: 0, 0, 0, 1,  9, 20, 13;\n  7: 0, 0, 0, 0,  4, 22, 38,  21;\n  8: 0, 0, 0, 0,  1, 14, 51,  71,  34;\n  9: 0, 0, 0, 0,  0,  5, 40, 111, 130, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + (x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (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] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (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}},{"oeis_id":"A128100","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  1;\n  2:  2,   1;\n  3:  3,   2;\n  4:  5,   5,   1;\n  5:  8,  10,   3;\n  6: 13,  20,   9,  1;\n  7: 21,  38,  22,  4;\n  8: 34,  71,  51, 14, 1;\n  9: 55, 130, 111, 40, 5;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + (x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + x^2)*P[n - 2, x]];\nTable[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = x*v[i - 1] + (x + x^2)*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(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":true,"trim_zeros":"low","regular_triangle":false}}],"derivation":{"root":"n-e7c079f81f452bf3","steps":[],"edges":[],"text":"A111006"},"headline":"A111006: Another version of Fibonacci-Pascal triangle A037027.","chain":null}