{"id":"n-5a6ab0e4ba52fc63","revision":1,"first_index":0,"fingerprint":"f6-797198ab3845c969","name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["576","1248","900","216"],["4608","13440","14688","7128","1296"],["36864","135168","198144","145152","53136","7776"],["294912","1302528","2396160","2350080","1296000","381024","46656"],["2359296","12189696","26984448","33177600","24468480","10824192","2659392","279936"],["18874368","111673344","289013760","427327488","394813440","233404416","86220288","18195840","1679616"],["150994944","1006632960","2982150144","5152702464","5722472448","4236115968","2090188800","662888448","122611968","10077696"],["1207959552","8959033344","29896998912","59114520576","76695994368","68223762432","42138206208","17844240384","4958226432","816293376","60466176"],["9663676416","78920024064","292930191360","652298158080","968255078400","1005966065664","746448224256","395583160320","146731253760","36279705600","5381489664","362797056"],["77309411328","689342251008","2816961675264","6975966412800","11659829575680","13857258995712","12007382188032","7643354628096","3547348992000","1170625167360","260730150912","35191314432","2176782336"],["618475290624","5978594476032","26671746908160","72709501353984","135134435082240","180817049419776","179202611478528","133191130152960","74238919704576","30649095290880","9109592211456","1845911420928","228562145280","13060694016"],["4947802324992","51539607552000","249245542121472","741706492280832","1517332488781824","2257343005851648","2518523188346880","2140744710094848","1393058138554368","690626280554496","256771309436928","69424844636160","12903965687808","1475858423808","78364164096"],["39582418599936","442003674365952","2303201982283776","7429125190975488","16588898863939584","27162738979504128","33692243541884928","32237096810840064","23988933369004032","13883359075762176","6197928158822400","2096026613710848","519780793319424","89230661517312","9482063855616","470184984576"],["316659348799488","3773523906527232","21077637904465920","73252213421506560","177285942057369600","316835305019670528","432514382212104192","460050235738030080","385334047817072640","255000472820121600","132883579725152256","53955781862621184","16734406028820480","3832530052055040","611240479948800","60653863010304","2821109907456"],["2533274790395904","32088147345014784","191262246674890752","712483534798848000","1857800816987996160","3598398092501581824","5361126887814856704","6275488179176865792","5842973796964761600","4352008069463408640","2593071474721947648","1228947733251883008","457609939406290944","131066676589363200","27885104151920640","4152673783775232","386492057321472","16926659444736"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["2940","41160","576240","8067360","112943040","1581202560","22136835840","309915701760","4338819824640","60743477544960","850408685629440","11905721598812160","166680102383370240","2333521433367183360","32669300067140567040"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["576","1248","900","216"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162893,"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-c0fedf7d6d9f66e7"]}],"region_exact":{},"incoming_edges":[{"id":"e-c0fedf7d6d9f66e7","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-5a6ab0e4ba52fc63","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":"a37985c9950c61d1d07045bc97ce19cc93d400bb266913c0d060f6d79bcd5b57","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c11cb6bca0e3e7dc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","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-c11cb6bca0e3e7dc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","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-36a383dc232988a5","kind":"oeis_match","node_a_id":"n-5a6ab0e4ba52fc63","node_b_id":null,"fingerprint":null,"oeis_id":"A168740","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":14},"in_catalog":false,"oeis_id":"A168740","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-623260e47c201702","kind":"oeis_match","node_a_id":"n-5a6ab0e4ba52fc63","node_b_id":null,"fingerprint":null,"oeis_id":"A168884","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":14},"in_catalog":false,"oeis_id":"A168884","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7d4f183e61c435dc","kind":"oeis_match","node_a_id":"n-5a6ab0e4ba52fc63","node_b_id":null,"fingerprint":null,"oeis_id":"A168932","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":14},"in_catalog":false,"oeis_id":"A168932","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-92e55e4cb0616e59","kind":"oeis_match","node_a_id":"n-5a6ab0e4ba52fc63","node_b_id":null,"fingerprint":null,"oeis_id":"A168788","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":14},"in_catalog":false,"oeis_id":"A168788","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-cd9edc0503117517","kind":"oeis_match","node_a_id":"n-5a6ab0e4ba52fc63","node_b_id":null,"fingerprint":null,"oeis_id":"A169028","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":14},"in_catalog":false,"oeis_id":"A169028","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d7ffaa45c563b3a1","kind":"oeis_match","node_a_id":"n-5a6ab0e4ba52fc63","node_b_id":null,"fingerprint":null,"oeis_id":"A168836","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":14},"in_catalog":false,"oeis_id":"A168836","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f7c61c15c5d8d0a7","kind":"oeis_match","node_a_id":"n-5a6ab0e4ba52fc63","node_b_id":null,"fingerprint":null,"oeis_id":"A168980","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":14},"in_catalog":false,"oeis_id":"A168980","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f9165e3cb74fec69","kind":"oeis_match","node_a_id":"n-5a6ab0e4ba52fc63","node_b_id":null,"fingerprint":null,"oeis_id":"A168692","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":14},"in_catalog":false,"oeis_id":"A168692","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A168692","relation":"match_row_sums"},{"oeis_id":"A168740","relation":"match_row_sums"},{"oeis_id":"A168788","relation":"match_row_sums"},{"oeis_id":"A168836","relation":"match_row_sums"},{"oeis_id":"A168884","relation":"match_row_sums"},{"oeis_id":"A168932","relation":"match_row_sums"},{"oeis_id":"A168980","relation":"match_row_sums"},{"oeis_id":"A169028","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:         576,         1248,           900,           216;\n  1:        4608,        13440,         14688,          7128,           1296;\n  2:       36864,       135168,        198144,        145152,          53136,           7776;\n  3:      294912,      1302528,       2396160,       2350080,        1296000,         381024,          46656;\n  4:     2359296,     12189696,      26984448,      33177600,       24468480,       10824192,        2659392,        279936;\n  5:    18874368,    111673344,     289013760,     427327488,      394813440,      233404416,       86220288,      18195840,       1679616;\n  6:   150994944,   1006632960,    2982150144,    5152702464,     5722472448,     4236115968,     2090188800,     662888448,     122611968,      10077696;\n  7:  1207959552,   8959033344,   29896998912,   59114520576,    76695994368,    68223762432,    42138206208,   17844240384,    4958226432,     816293376,     60466176;\n  8:  9663676416,  78920024064,  292930191360,  652298158080,   968255078400,  1005966065664,   746448224256,  395583160320,  146731253760,   36279705600,   5381489664,   362797056;\n  9: 77309411328, 689342251008, 2816961675264, 6975966412800, 11659829575680, 13857258995712, 12007382188032, 7643354628096, 3547348992000, 1170625167360, 260730150912, 35191314432, 2176782336;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 1248*x + 900*x^2 + 216*x^3.","mathematica":"P[0, x_] := 576 + 1248*x + 900*x^2 + 216*x^3;\nP[n_, x_] := P[n, x] = Expand[(8 + 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] = 576 + 1248*x + 900*x^2 + 216*x^3; for(n = 1, N, my(i = n + 1); v[i] = (8 + 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-c11cb6bca0e3e7dc","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-c0fedf7d6d9f66e7"],"text":"A038284 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A038284 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
