{"id":"n-7d37bf0e0a7dbf4f","revision":1,"first_index":0,"fingerprint":"f6-97b5915faaad55c8","name":"obreschkoff_combination of Triangle of coefficients in expansion of (1+6x)^n.","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","6"],["2","18","36"],["2","30","144","216"],["2","42","324","1080","1296"],["2","54","576","3024","7776","7776"],["2","66","900","6480","25920","54432","46656"],["2","78","1296","11880","64800","209952","373248","279936"],["2","90","1764","19656","136080","598752","1632960","2519424","1679616"],["2","102","2304","30240","254016","1415232","5225472","12317184","16796160","10077696"],["2","114","2916","44064","435456","2939328","13716864","43670016","90699264","110854656","60466176"],["2","126","3600","61560","699840","5552064","31352832","125971200","352719360","655050240","725594112","362797056"],["2","138","4356","83160","1069200","9751104","64665216","314088192","1108546560","2771366400","4655895552","4716361728","2176782336"],["2","150","5184","109296","1568160","16166304","123171840","702079488","2993075712","9422645760","21284093952","32651735040","30474952704","13060694016"],["2","162","6084","140400","2223936","25575264","220169664","1441110528","7205552640","27381100032","77819968512","160356298752","226385362944","195910410240","78364164096"],["2","174","7056","176904","3066336","38918880","373621248","2762128512","15852215808","70614415872","242106568704","627276109824","1188523155456","1554222587904","1253826625536","470184984576"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["8","56","392","2744","19208","134456","941192","6588344","46118408","322828856","2259801992","15818613944","110730297608","775112083256","5425784582792"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162781,"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-84f0fe7fd2f0b117"]}],"region_exact":{},"incoming_edges":[{"id":"e-84f0fe7fd2f0b117","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-7d37bf0e0a7dbf4f","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":"ee8e16c459ffa1f1ce4cda3ab7d55aad22a739f685353c0cecb7d0006f3fcb16","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-efce1b6c07b70e1a","role":"sequence","name":"Triangle of coefficients in expansion of (1+6x)^n.","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-efce1b6c07b70e1a","role":"sequence","name":"Triangle of coefficients in expansion of (1+6x)^n.","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-013ff884f7eefe9a","kind":"oeis_match","node_a_id":"n-7d37bf0e0a7dbf4f","node_b_id":null,"fingerprint":null,"oeis_id":"A167899","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":"A167899","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2407c03520dbaab4","kind":"oeis_match","node_a_id":"n-7d37bf0e0a7dbf4f","node_b_id":null,"fingerprint":null,"oeis_id":"A168733","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":"A168733","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2f7441ace4b5c9ab","kind":"oeis_match","node_a_id":"n-7d37bf0e0a7dbf4f","node_b_id":null,"fingerprint":null,"oeis_id":"A168781","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":"A168781","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-59778a08b6fabefe","kind":"oeis_match","node_a_id":"n-7d37bf0e0a7dbf4f","node_b_id":null,"fingerprint":null,"oeis_id":"A168685","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":"A168685","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-65c95b4c57af2bfb","kind":"oeis_match","node_a_id":"n-7d37bf0e0a7dbf4f","node_b_id":null,"fingerprint":null,"oeis_id":"A168877","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":"A168877","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-73cd553f9d3b80fe","kind":"oeis_match","node_a_id":"n-7d37bf0e0a7dbf4f","node_b_id":null,"fingerprint":null,"oeis_id":"A168829","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":"A168829","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e0f2073017a95ccf","kind":"oeis_match","node_a_id":"n-7d37bf0e0a7dbf4f","node_b_id":null,"fingerprint":null,"oeis_id":"A168925","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":"A168925","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e5f8899127e7cf9b","kind":"oeis_match","node_a_id":"n-7d37bf0e0a7dbf4f","node_b_id":null,"fingerprint":null,"oeis_id":"A003950","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":"A003950","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003950","relation":"match_row_sums"},{"oeis_id":"A167899","relation":"match_row_sums"},{"oeis_id":"A168685","relation":"match_row_sums"},{"oeis_id":"A168733","relation":"match_row_sums"},{"oeis_id":"A168781","relation":"match_row_sums"},{"oeis_id":"A168829","relation":"match_row_sums"},{"oeis_id":"A168877","relation":"match_row_sums"},{"oeis_id":"A168925","relation":"match_row_sums"}],"bfiles":[],"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,   6;\n  1: 2,  18,   36;\n  2: 2,  30,  144,   216;\n  3: 2,  42,  324,  1080,   1296;\n  4: 2,  54,  576,  3024,   7776,    7776;\n  5: 2,  66,  900,  6480,  25920,   54432,    46656;\n  6: 2,  78, 1296, 11880,  64800,  209952,   373248,   279936;\n  7: 2,  90, 1764, 19656, 136080,  598752,  1632960,  2519424,  1679616;\n  8: 2, 102, 2304, 30240, 254016, 1415232,  5225472, 12317184, 16796160,  10077696;\n  9: 2, 114, 2916, 44064, 435456, 2939328, 13716864, 43670016, 90699264, 110854656, 60466176;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 2 + 6*x.","mathematica":"P[0, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 6*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 + 6*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 6*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-efce1b6c07b70e1a","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-84f0fe7fd2f0b117"],"text":"A013613 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A013613 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
