{"id":"n-a202ff64d2ae7451","revision":1,"first_index":0,"fingerprint":"f6-f8a3482d25b550b7","name":"obreschkoff_combination of Cube of lower triangular normalized binomial matrix.","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","1"],["12","7","1"],["36","33","10","1"],["108","135","63","13","1"],["324","513","324","102","16","1"],["972","1863","1485","630","150","19","1"],["2916","6561","6318","3375","1080","207","22","1"],["8748","22599","25515","16443","6615","1701","273","25","1"],["26244","76545","99144","74844","36288","11718","2520","348","28","1"],["78732","255879","373977","323676","183708","71442","19278","3564","432","31","1"],["236196","846369","1377810","1345005","874800","398034","129276","29970","4860","525","34","1"],["708588","2775303","4979799","5412825","3969405","2068902","785862","219186","44550","6435","627","37","1"],["2125764","9034497","17714700","21218274","17321040","10176111","4426488","1443420","352836","63855","8316","738","40","1"],["6377292","29229255","62178597","81369522","73181394","47849373","23455575","8756748","2501928","544401","88803","10530","858","43","1"],["19131876","94065057","215765046","306287163","300913704","216729513","118216098","49725819","16262532","4135131","810810","120393","13104","987","46","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","20","80","320","1280","5120","20480","81920","327680","1310720","5242880","20971520","83886080","335544320","1342177280"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162723,"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-30d7db2c4a1b8418"]}],"region_exact":{},"incoming_edges":[{"id":"e-30d7db2c4a1b8418","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-a202ff64d2ae7451","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":"163876e8f8b69a91c9dcdbe11ff518f47751f8957afea81fdb9fe2889efa308b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-8d887bfc40d036b9","role":"sequence","name":"Cube of lower triangular normalized binomial matrix.","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-8d887bfc40d036b9","role":"sequence","name":"Cube of lower triangular normalized binomial matrix.","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-0d77e9b30d800b0d","kind":"oeis_match","node_a_id":"n-a202ff64d2ae7451","node_b_id":null,"fingerprint":null,"oeis_id":"A003947","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":"A003947","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-13c56e9abba65fab","kind":"oeis_match","node_a_id":"n-a202ff64d2ae7451","node_b_id":null,"fingerprint":null,"oeis_id":"A168826","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":"A168826","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4ef593c1df2f880d","kind":"oeis_match","node_a_id":"n-a202ff64d2ae7451","node_b_id":null,"fingerprint":null,"oeis_id":"A168778","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":"A168778","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7cacae85342eb941","kind":"oeis_match","node_a_id":"n-a202ff64d2ae7451","node_b_id":null,"fingerprint":null,"oeis_id":"A168730","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":"A168730","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a00b2b66fa8bd2fe","kind":"oeis_match","node_a_id":"n-a202ff64d2ae7451","node_b_id":null,"fingerprint":null,"oeis_id":"A168874","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":"A168874","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d3d69735a7249d5a","kind":"oeis_match","node_a_id":"n-a202ff64d2ae7451","node_b_id":null,"fingerprint":null,"oeis_id":"A168922","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":"A168922","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e7f1e51ffd6f7263","kind":"oeis_match","node_a_id":"n-a202ff64d2ae7451","node_b_id":null,"fingerprint":null,"oeis_id":"A168682","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":"A168682","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fe8bd48a8e54193f","kind":"oeis_match","node_a_id":"n-a202ff64d2ae7451","node_b_id":null,"fingerprint":null,"oeis_id":"A167896","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":"A167896","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003947","relation":"match_row_sums"},{"oeis_id":"A167896","relation":"match_row_sums"},{"oeis_id":"A168682","relation":"match_row_sums"},{"oeis_id":"A168730","relation":"match_row_sums"},{"oeis_id":"A168778","relation":"match_row_sums"},{"oeis_id":"A168826","relation":"match_row_sums"},{"oeis_id":"A168874","relation":"match_row_sums"},{"oeis_id":"A168922","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:     4,      1;\n  1:    12,      7,      1;\n  2:    36,     33,     10,      1;\n  3:   108,    135,     63,     13,      1;\n  4:   324,    513,    324,    102,     16,     1;\n  5:   972,   1863,   1485,    630,    150,    19,     1;\n  6:  2916,   6561,   6318,   3375,   1080,   207,    22,    1;\n  7:  8748,  22599,  25515,  16443,   6615,  1701,   273,   25,   1;\n  8: 26244,  76545,  99144,  74844,  36288, 11718,  2520,  348,  28,  1;\n  9: 78732, 255879, 373977, 323676, 183708, 71442, 19278, 3564, 432, 31, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + x.","mathematica":"P[0, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[(3 + 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 + x; for(n = 1, N, my(i = n + 1); v[i] = (3 + 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-8d887bfc40d036b9","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-30d7db2c4a1b8418"],"text":"A027465 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A027465 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}