{"id":"n-0d55e0bb3c384646","revision":2,"first_index":0,"fingerprint":"f6-00788a960ffaf339","name":"Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).","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":false,"rows":[["1"],["1"],["1"],["1","1"],["1","2"],["1","3"],["1","4","1"],["1","5","3"],["1","6","6"],["1","7","10","1"],["1","8","15","4"],["1","9","21","10"],["1","10","28","20","1"],["1","11","36","35","5"],["1","12","45","56","15"]],"degrees":[0,0,0,1,1,1,2,2,2,3,3,3,4,4,4],"row_sums":["1","1","1","2","3","4","6","9","13","19","28","41","60","88","129"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A102547","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:reversal","source_kind":"derived","equality_edge_id":"e-9288ac896b9000e4","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159005,"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-9cfbf7a52f4ba322","e-eef18680cd27a7c9"]},{"id":159006,"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-9cfbf7a52f4ba322","e-eef18680cd27a7c9"]},{"id":159007,"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-9cfbf7a52f4ba322","e-eef18680cd27a7c9"]},{"id":159008,"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-eef18680cd27a7c9"]}],"region_exact":{},"incoming_edges":[{"id":"e-9288ac896b9000e4","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-0d55e0bb3c384646","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"8be6c1649d86be581a6237945cf2a81f00fd046f047a037eaefcc2539e34df96","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-eef18680cd27a7c9","operation":{"operation":"parent_provers","route":"paper_theorem"},"operation_detail":"paper_theorem","conclusion_id":"n-0d55e0bb3c384646","premise_count":0,"theorem":"polytool parent provers, route paper_theorem: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"9fce553613c84641e95a082c164cd6e4538ea7e88452b5ff525692603f95b6a7","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":[]},{"id":"e-9cfbf7a52f4ba322","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":1,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-0d55e0bb3c384646","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"3adeeac61cf9a6a0d93ca8f2420bffea3c23ab8f548151288439f442afde6c62","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-67a75673619cc276","role":"sequence","name":"Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.","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":[{"id":"e-2461a7f1545a7610","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-9e4c6a8ebe739ee2","conclusion_name":"basis_map_A008277 of Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318)."},{"id":"e-82b4d7baf7961b16","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-28041dfac2a3c4ec","conclusion_name":"basis_map_A013609 of Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318)."},{"id":"e-ac961eccb8b6d01b","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-6e06ff1c41cd6465","conclusion_name":"shift_x_plus_one of Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318)."},{"id":"e-f413e745ff3e72cb","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-c7c6fb30e7a6c763","conclusion_name":"basis_map_A013610 of Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318)."},{"id":"e-f9ff31509ec6911f","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-04616b0b8f6da9ca","conclusion_name":"basis_map_A013611 of Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318)."},{"id":"e-0e9e334768ea0a35","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-67a75673619cc276","conclusion_name":"Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1."},{"id":"e-726c0b03589eafe2","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-512e5935bc8384f1","conclusion_name":"Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 001 (n,k>=0)."},{"id":"e-8e4f8cf0a7ba3e7c","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-6e06ff1c41cd6465","conclusion_name":"shift_x_plus_one of Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318)."},{"id":"e-b8d37b5fe37403cf","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-a5b33928f20c14a2","conclusion_name":"Triangle T(n,k) = 3*T(n-1,k) +  T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows."}],"used_by_total":9,"links":[{"id":"l-79545f49d9cfc0e4","kind":"fingerprint_related","node_a_id":"n-0d55e0bb3c384646","node_b_id":"n-67a75673619cc276","fingerprint":"f6-00788a960ffaf339","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-00788a960ffaf339"},"other_id":"n-67a75673619cc276","other":{"id":"n-67a75673619cc276","name":"Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.","first_index":0,"fingerprint":"f6-00788a960ffaf339","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["0","1"],["0","0","1"],["0","0","0","1"]],"card_rows":[["0","1"],["0","0","1"],["0","0","0","1"],["0","0","0","1","1"]],"row_sums_preview":["1","1","1","2","3","4","6","9"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d55e0bb3c384646","steps":["reverse"],"edges":["e-0e9e334768ea0a35"],"text":"A102547 \u2192 reverse"},"headline":"A102547 \u2192 reverse","chain":null}},{"id":"l-0224dcf894bd2859","kind":"oeis_match","node_a_id":"n-0d55e0bb3c384646","node_b_id":null,"fingerprint":null,"oeis_id":"A102547","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":15,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":15,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A102547","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-58c7e138dfb1cd41","kind":"oeis_match","node_a_id":"n-0d55e0bb3c384646","node_b_id":null,"fingerprint":null,"oeis_id":"A000930","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":"A000930","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a892cf8a9223823e","kind":"oeis_match","node_a_id":"n-0d55e0bb3c384646","node_b_id":null,"fingerprint":null,"oeis_id":"A078012","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A078012","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ff354fbe141fa873","kind":"oeis_match","node_a_id":"n-0d55e0bb3c384646","node_b_id":null,"fingerprint":null,"oeis_id":"A068921","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":"A068921","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A102547","relation":"identification"},{"oeis_id":"A000930","relation":"match_row_sums"},{"oeis_id":"A068921","relation":"match_row_sums"},{"oeis_id":"A078012","relation":"match_row_sums"},{"oeis_id":"A102547","relation":"match_triangle"}],"bfiles":[{"node_id":"n-0d55e0bb3c384646","oeis_id":"A000930","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":44,"oeis_terms_available":44,"term_count":200,"raw_size":4255},{"node_id":"n-0d55e0bb3c384646","oeis_id":"A068921","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":42,"oeis_terms_available":42,"term_count":200,"raw_size":4287},{"node_id":"n-0d55e0bb3c384646","oeis_id":"A102547","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":76,"terms_matched":45,"oeis_terms_agreeing":80,"oeis_terms_available":80,"term_count":1001,"raw_size":10499}],"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: 1, 1;\n  4: 1, 2;\n  5: 1, 3;\n  6: 1, 4,  1;\n  7: 1, 5,  3;\n  8: 1, 6,  6;\n  9: 1, 7, 10, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-1,x) + 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[P[n - 1, x] + 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] = v[i - 1] + 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}},{"oeis_id":"A102547","target":"triangle","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: 1, 1;\n  4: 1, 2;\n  5: 1, 3;\n  6: 1, 4,  1;\n  7: 1, 5,  3;\n  8: 1, 6,  6;\n  9: 1, 7, 10, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-1,x) + 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[P[n - 1, x] + 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] = v[i - 1] + 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-0d55e0bb3c384646","steps":[],"edges":[],"text":"A102547"},"headline":"A102547: Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).","chain":null}