{"id":"n-86fcb9f018178f39","revision":1,"first_index":0,"fingerprint":"f6-efce8cd29f4f1fd0","name":"obreschkoff_combination of binomial","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":true,"rows":[["2","1"],["2","3","1"],["2","5","4","1"],["2","7","9","5","1"],["2","9","16","14","6","1"],["2","11","25","30","20","7","1"],["2","13","36","55","50","27","8","1"],["2","15","49","91","105","77","35","9","1"],["2","17","64","140","196","182","112","44","10","1"],["2","19","81","204","336","378","294","156","54","11","1"],["2","21","100","285","540","714","672","450","210","65","12","1"],["2","23","121","385","825","1254","1386","1122","660","275","77","13","1"],["2","25","144","506","1210","2079","2640","2508","1782","935","352","90","14","1"],["2","27","169","650","1716","3289","4719","5148","4290","2717","1287","442","104","15","1"],["2","29","196","819","2366","5005","8008","9867","9438","7007","4004","1729","546","119","16","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","6","12","24","48","96","192","384","768","1536","3072","6144","12288","24576","49152"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162702,"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-c4dc09241b591172"]}],"region_exact":{},"incoming_edges":[{"id":"e-c4dc09241b591172","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-86fcb9f018178f39","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"90093b250e1dc76d9c702d0e66bda53fa24ce49009b8e062f70187cc53f64789","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c8c19eba9d68881f","role":"sequence","name":"binomial","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-c8c19eba9d68881f","role":"sequence","name":"binomial","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-d11c54a565224b64","kind":"fingerprint_related","node_a_id":"n-86fcb9f018178f39","node_b_id":"n-2ab5cee8137b2478","fingerprint":"f6-efce8cd29f4f1fd0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-efce8cd29f4f1fd0"},"other_id":"n-2ab5cee8137b2478","other":{"id":"n-2ab5cee8137b2478","name":"obreschkoff_combination of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-efce8cd29f4f1fd0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","2"],["4","6","2"],["4","10","8","2"]],"card_rows":[["4","2"],["4","6","2"],["4","10","8","2"],["4","14","18","10","2"]],"row_sums_preview":["6","12","24","48","96","192","384","768"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0787c477c17aca9a"],"text":"A028326 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028326 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}},{"id":"l-0a3066d7ac1fe527","kind":"oeis_match","node_a_id":"n-86fcb9f018178f39","node_b_id":null,"fingerprint":null,"oeis_id":"A122391","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":"A122391","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-1a0c6929da366cc7","kind":"oeis_match","node_a_id":"n-86fcb9f018178f39","node_b_id":null,"fingerprint":null,"oeis_id":"A168728","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":"A168728","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-228c96fa31bfa866","kind":"oeis_match","node_a_id":"n-86fcb9f018178f39","node_b_id":null,"fingerprint":null,"oeis_id":"A167881","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":"A167881","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-402411712a2f628a","kind":"oeis_match","node_a_id":"n-86fcb9f018178f39","node_b_id":null,"fingerprint":null,"oeis_id":"A003945","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":"A003945","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4c3364d0b3267815","kind":"oeis_match","node_a_id":"n-86fcb9f018178f39","node_b_id":null,"fingerprint":null,"oeis_id":"A098011","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A098011","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-638b1030994a4cd7","kind":"oeis_match","node_a_id":"n-86fcb9f018178f39","node_b_id":null,"fingerprint":null,"oeis_id":"A042950","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":"A042950","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6bf139a194f4f9c0","kind":"oeis_match","node_a_id":"n-86fcb9f018178f39","node_b_id":null,"fingerprint":null,"oeis_id":"A168680","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":"A168680","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bb7ecd460715269c","kind":"oeis_match","node_a_id":"n-86fcb9f018178f39","node_b_id":null,"fingerprint":null,"oeis_id":"A007283","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":"A007283","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003945","relation":"match_row_sums"},{"oeis_id":"A007283","relation":"match_row_sums"},{"oeis_id":"A042950","relation":"match_row_sums"},{"oeis_id":"A098011","relation":"match_row_sums"},{"oeis_id":"A122391","relation":"match_row_sums"},{"oeis_id":"A167881","relation":"match_row_sums"},{"oeis_id":"A168680","relation":"match_row_sums"},{"oeis_id":"A168728","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-86fcb9f018178f39","oeis_id":"A007283","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":33,"oeis_terms_available":33,"term_count":200,"raw_size":7076}],"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,  1;\n  1: 2,  3,  1;\n  2: 2,  5,  4,   1;\n  3: 2,  7,  9,   5,   1;\n  4: 2,  9, 16,  14,   6,   1;\n  5: 2, 11, 25,  30,  20,   7,   1;\n  6: 2, 13, 36,  55,  50,  27,   8,   1;\n  7: 2, 15, 49,  91, 105,  77,  35,   9,  1;\n  8: 2, 17, 64, 140, 196, 182, 112,  44, 10,  1;\n  9: 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1]); 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-c8c19eba9d68881f","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-c4dc09241b591172"],"text":"A007318 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A007318 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}