{"id":"n-2422ccdf8fc66e9a","revision":1,"first_index":0,"fingerprint":"f6-3c4a6b4bc3495597","name":"basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","12","4"],["36","144","192","96","16"],["216","1296","3024","3456","2016","576","64"],["1296","10368","34560","62208","65664","41472","15360","3072","256"],["7776","77760","336960","829440","1278720","1285632","852480","368640","99840","15360","1024"],["46656","559872","2985984","9331200","18973440","26376192","25657344","17584128","8432640","2764800","589824","73728","4096"],["279936","3919104","24820992","94058496","237758976","423263232","546352128","518897664","364234752","188116992","70447104","18579456","3268608","344064","16384"],["1679616","26873856","197074944","877879296","2654539776","5768921088","9308307456","11362664448","10597588992","7575109632","4137025536","1709309952","524353536","115605504","17301504","1572864","65536"],["10077696","181398528","1511654400","7739670528","27250089984","69979521024","135695056896","202951360512","237170737152","218072383488","158113824768","90200604672","40205942784","13823115264","3588489216","679477248","88473600","7077888","262144"],["60466176","1209323520","11287019520","65303470080","262423203840","777836888064","1762924953600","3125966929920","4401220976640","4966288588800","4514234499072","3310859059200","1956098211840","926212423680","348232089600","102431195136","23038525440","3822059520","440401920","31457280","1048576"],["362797056","7981535232","82475864064","532102348800","2403328942080","8077313654784","20961285193728","43022248574976","70970628833280","95116250972160","104285753966592","93901122699264","69523835977728","42273889320960","21028334469120","8498221940736","2760333852672","709119442944","140661227520","20761804800","2145386496","138412032","4194304"],["2176782336","52242776064","592084795392","4214250602496","21135105294336","79432238628864","232308790788096","541978168393728","1025935896674304","1594434046132224","2050992050798592","2195300787683328","1961099504123904","1463533858455552","911552022577152","472424902557696","202654004281344","71371610652672","20394736091136","4648983330816","824658886656","109622329344","10267656192","603979776","16777216"],["13060694016","339578044416","4188129214464","32599492263936","179749978177536","747071697715200","2431580029452288","6357303454334976","13588588563922944","24045747710459904","35542904445075456","44161445520211968","46314174680137728","41095600350953472","30876116453425152","19627309120094208","10531230946689024","4749777325522944","1789443761504256","558117175689216","142314697654272","29149573939200","4675709435904","565325070336","48419045376","2617245696","67108864"],["78364164096","2194196594688","29255954595840","247212816334848","1486446293090304","6769827893477376","24273340462006272","70311067870298112","167545492953366528","332766762847174656","556160753451663360","787666517304016896","949994012103671808","978989480348221440","863660601652543488","652659653565480960","422219560934965248","233382671793782784","109858914262056960","43821137494278144","14709069340213248","4115142518243328","947107934502912","176098759409664","25777252859904","2858032300032","225485783040","11274289152","268435456"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","22","484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792","26559922791424","584318301411328","12855002631049216","282810057883082752","6221821273427820544"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","12","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166677,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-e79851ce4a3e511f"]},{"id":166678,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-e79851ce4a3e511f"]},{"id":166679,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-e79851ce4a3e511f"]}],"region_exact":{},"incoming_edges":[{"id":"e-e79851ce4a3e511f","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-2422ccdf8fc66e9a","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e19968ffc5d0a4f00de6bf92841ece4d8d200c12f52b4e8c75eb92e2eece444a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d3c3090977105807","role":"sequence","name":"Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-a142fa84029060a1","kind":"fingerprint_related","node_a_id":"n-2422ccdf8fc66e9a","node_b_id":"n-c4fd821a14172d5f","fingerprint":"f6-3c4a6b4bc3495597","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3c4a6b4bc3495597"},"other_id":"n-c4fd821a14172d5f","other":{"id":"n-c4fd821a14172d5f","name":"basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","first_index":0,"fingerprint":"f6-3c4a6b4bc3495597","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","18","9"],["36","216","432","324","81"]],"card_rows":[["1"],["6","18","9"],["36","216","432","324","81"],["216","1944","6804","11664","10206","4374","729"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d3c3090977105807","steps":["basis A013610"],"edges":["e-0f2bf5edb3121755"],"text":"A272867 \u2192 basis A013610"},"headline":"A272867 \u2192 basis A013610","chain":null}},{"id":"l-cfcd4850fab95987","kind":"oeis_match","node_a_id":"n-2422ccdf8fc66e9a","node_b_id":null,"fingerprint":null,"oeis_id":"A009966","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":"A009966","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009966","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-2422ccdf8fc66e9a","oeis_id":"A009966","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":27706}],"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: 1;\n 1: 6, 12, 4;\n 2: 36, 144, 192, 96, 16;\n 3: 216, 1296, 3024, 3456, 2016, 576, 64;\n 4: 1296, 10368, 34560, 62208, 65664, 41472, 15360, 3072, 256;\n 5: 7776, 77760, 336960, 829440, 1278720, 1285632, 852480, 368640, 99840, 15360, 1024;\n 6: 46656, 559872, 2985984, 9331200, 18973440, 26376192, 25657344, 17584128, 8432640, 2764800, 589824, 73728, 4096;\n 7: 279936, 3919104, 24820992, 94058496, 237758976, 423263232, 546352128, 518897664, 364234752, 188116992, 70447104, 18579456, 3268608, 344064, 16384;\n 8: 1679616, 26873856, 197074944, 877879296, 2654539776, 5768921088, 9308307456, 11362664448, 10597588992, 7575109632, 4137025536, 1709309952, 524353536, 115605504, 17301504, 1572864, 65536;\n 9: 10077696, 181398528, 1511654400, 7739670528, 27250089984, 69979521024, 135695056896, 202951360512, 237170737152, 218072383488, 158113824768, 90200604672, 40205942784, 13823115264, 3588489216, 679477248, 88473600, 7077888, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+12*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + 12*x + 4*x^2)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6 + 12*x + 4*x^2)*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-d3c3090977105807","steps":["basis A013609"],"edges":["e-e79851ce4a3e511f"],"text":"A272867 \u2192 basis A013609"},"headline":"A272867 \u2192 basis A013609","chain":null}