{"id":"n-2ab5cee8137b2478","revision":1,"first_index":0,"fingerprint":"f6-efce8cd29f4f1fd0","name":"obreschkoff_combination of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","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":[["4","2"],["4","6","2"],["4","10","8","2"],["4","14","18","10","2"],["4","18","32","28","12","2"],["4","22","50","60","40","14","2"],["4","26","72","110","100","54","16","2"],["4","30","98","182","210","154","70","18","2"],["4","34","128","280","392","364","224","88","20","2"],["4","38","162","408","672","756","588","312","108","22","2"],["4","42","200","570","1080","1428","1344","900","420","130","24","2"],["4","46","242","770","1650","2508","2772","2244","1320","550","154","26","2"],["4","50","288","1012","2420","4158","5280","5016","3564","1870","704","180","28","2"],["4","54","338","1300","3432","6578","9438","10296","8580","5434","2574","884","208","30","2"],["4","58","392","1638","4732","10010","16016","19734","18876","14014","8008","3458","1092","238","32","2"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["6","12","24","48","96","192","384","768","1536","3072","6144","12288","24576","49152","98304"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162703,"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-0787c477c17aca9a"]}],"region_exact":{},"incoming_edges":[{"id":"e-0787c477c17aca9a","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-2ab5cee8137b2478","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":"e02cd713f73c0a8517fab695efce4cc901e15fbff0a9b3d5baa7805c6b3805fd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-089a9b4f824a1c20","role":"sequence","name":"Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","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-089a9b4f824a1c20","role":"sequence","name":"Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","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-86fcb9f018178f39","other":{"id":"n-86fcb9f018178f39","name":"obreschkoff_combination of binomial","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":[["2","1"],["2","3","1"],["2","5","4","1"]],"card_rows":[["2","1"],["2","3","1"],["2","5","4","1"],["2","7","9","5","1"]],"row_sums_preview":["3","6","12","24","48","96","192","384"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-223146b284f77de0","kind":"oeis_match","node_a_id":"n-2ab5cee8137b2478","node_b_id":null,"fingerprint":null,"oeis_id":"A111286","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":"A111286","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-54a21385edd59995","kind":"oeis_match","node_a_id":"n-2ab5cee8137b2478","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168680","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-634d410ffa860a82","kind":"oeis_match","node_a_id":"n-2ab5cee8137b2478","node_b_id":null,"fingerprint":null,"oeis_id":"A058764","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A058764","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-69515791b0e9758f","kind":"oeis_match","node_a_id":"n-2ab5cee8137b2478","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007283","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6a12bbbf1b06d4fc","kind":"oeis_match","node_a_id":"n-2ab5cee8137b2478","node_b_id":null,"fingerprint":null,"oeis_id":"A091629","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A091629","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9dbb4daa4e3c293e","kind":"oeis_match","node_a_id":"n-2ab5cee8137b2478","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A042950","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e169f6bbc25ff450","kind":"oeis_match","node_a_id":"n-2ab5cee8137b2478","node_b_id":null,"fingerprint":null,"oeis_id":"A082505","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A082505","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e1a16e73114986d3","kind":"oeis_match","node_a_id":"n-2ab5cee8137b2478","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A003945","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":"A058764","relation":"match_row_sums"},{"oeis_id":"A082505","relation":"match_row_sums"},{"oeis_id":"A091629","relation":"match_row_sums"},{"oeis_id":"A111286","relation":"match_row_sums"},{"oeis_id":"A168680","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-2ab5cee8137b2478","oeis_id":"A091629","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":7138}],"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: 4,  2;\n  1: 4,  6,   2;\n  2: 4, 10,   8,   2;\n  3: 4, 14,  18,  10,   2;\n  4: 4, 18,  32,  28,  12,   2;\n  5: 4, 22,  50,  60,  40,  14,   2;\n  6: 4, 26,  72, 110, 100,  54,  16,   2;\n  7: 4, 30,  98, 182, 210, 154,  70,  18,   2;\n  8: 4, 34, 128, 280, 392, 364, 224,  88,  20,  2;\n  9: 4, 38, 162, 408, 672, 756, 588, 312, 108, 22, 2;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 2*x.","mathematica":"P[0, x_] := 4 + 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] = 4 + 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-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}