{"id":"n-8a3a4358b42999f7","revision":1,"first_index":0,"fingerprint":"f6-4adf7795e7695c39","name":"obreschkoff_combination of Triangle of coefficients in expansion of (1+11x)^n.","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":[["2","55","484","1331"],["2","77","1089","6655","14641"],["2","99","1936","18634","87846","161051"],["2","121","3025","39930","292820","1127357","1771561"],["2","143","4356","73205","732050","4348377","14172488","19487171"],["2","165","5929","121121","1537305","12400927","62004635","175384539","214358881"],["2","187","7744","186340","2869636","29311282","198414832","857435524","2143588810","2357947691"],["2","209","9801","271524","4919376","60877278","520838934","3039998676","11575379574","25937424601","25937424601"],["2","231","12100","379335","7906140","114990414","1190488992","8769226950","45015365010","153266599915","311249095212","285311670611"],["2","253","14641","512435","12078825","201957954","2455383546","21864605862","141476861460","648435615025","1997181694277","3709051717943","3138428376721"],["2","275","17424","673486","17715610","334825029","4676921040","48873824868","381987525942","2204681091085","9129973459552","25678050354990","43937997274094","34522712143931"],["2","297","20449","865150","25123956","529696739","8359996359","100319956308","919599599490","6406543876447","33381465461487","126107758410062","326396551178984","517840682158965","379749833583241"],["2","319","23716","1090089","34640606","806060255","14186660488","192279916257","2023119118878","16522139470837","103853448102404","493303878486419","1713581893689666","4108202745127789","6075997337331856","4177248169415651"],["2","341","27225","1350965","46631585","1187106921","23053323293","348333181625","4138198197705","38776449778495","285596982281611","1635691807612863","7139924557040275","22957603575714115","51266227533737535","71013218880066067","45949729863572161"],["2","363","30976","1650440","61492200","1700054356","36111499424","601919737848","7969863195580","84296629953250","712137929845056","4777258612710584","25132534440781768","101496773703157140","303799866866592800","634941721751178952","827095137544298898","505447028499293771"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["1872","22464","269568","3234816","38817792","465813504","5589762048","67077144576","804925734912","9659108818944","115909305827328","1390911669927936","16690940039135232","200291280469622784","2403495365635473408"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","55","484","1331"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162883,"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-f27f365b1aa97cdc"]}],"region_exact":{},"incoming_edges":[{"id":"e-f27f365b1aa97cdc","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-8a3a4358b42999f7","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":"1e52962be8f1283a486899ba2f3e98ea333f27eed9531168047c068e9b3fd0fc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-b1e393c17b0898c2","role":"sequence","name":"Triangle of coefficients in expansion of (1+11x)^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-b1e393c17b0898c2","role":"sequence","name":"Triangle of coefficients in expansion of (1+11x)^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-03911b2b9417da52","kind":"oeis_match","node_a_id":"n-8a3a4358b42999f7","node_b_id":null,"fingerprint":null,"oeis_id":"A168690","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":"A168690","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-45f22adafb4be9cf","kind":"oeis_match","node_a_id":"n-8a3a4358b42999f7","node_b_id":null,"fingerprint":null,"oeis_id":"A168978","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":"A168978","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6ade938f8585d176","kind":"oeis_match","node_a_id":"n-8a3a4358b42999f7","node_b_id":null,"fingerprint":null,"oeis_id":"A168834","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":"A168834","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8150f578eb6e8863","kind":"oeis_match","node_a_id":"n-8a3a4358b42999f7","node_b_id":null,"fingerprint":null,"oeis_id":"A168930","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":"A168930","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c878d00a3a786529","kind":"oeis_match","node_a_id":"n-8a3a4358b42999f7","node_b_id":null,"fingerprint":null,"oeis_id":"A168738","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":"A168738","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d7cbf88729f8067c","kind":"oeis_match","node_a_id":"n-8a3a4358b42999f7","node_b_id":null,"fingerprint":null,"oeis_id":"A169026","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":"A169026","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-eaaaaadc1d0cd419","kind":"oeis_match","node_a_id":"n-8a3a4358b42999f7","node_b_id":null,"fingerprint":null,"oeis_id":"A168882","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":"A168882","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fc3f38bdc229567a","kind":"oeis_match","node_a_id":"n-8a3a4358b42999f7","node_b_id":null,"fingerprint":null,"oeis_id":"A168786","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":"A168786","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A168690","relation":"match_row_sums"},{"oeis_id":"A168738","relation":"match_row_sums"},{"oeis_id":"A168786","relation":"match_row_sums"},{"oeis_id":"A168834","relation":"match_row_sums"},{"oeis_id":"A168882","relation":"match_row_sums"},{"oeis_id":"A168930","relation":"match_row_sums"},{"oeis_id":"A168978","relation":"match_row_sums"},{"oeis_id":"A169026","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,  55,   484,   1331;\n  1: 2,  77,  1089,   6655,    14641;\n  2: 2,  99,  1936,  18634,    87846,    161051;\n  3: 2, 121,  3025,  39930,   292820,   1127357,    1771561;\n  4: 2, 143,  4356,  73205,   732050,   4348377,   14172488,    19487171;\n  5: 2, 165,  5929, 121121,  1537305,  12400927,   62004635,   175384539,    214358881;\n  6: 2, 187,  7744, 186340,  2869636,  29311282,  198414832,   857435524,   2143588810,   2357947691;\n  7: 2, 209,  9801, 271524,  4919376,  60877278,  520838934,  3039998676,  11575379574,  25937424601,   25937424601;\n  8: 2, 231, 12100, 379335,  7906140, 114990414, 1190488992,  8769226950,  45015365010, 153266599915,  311249095212,  285311670611;\n  9: 2, 253, 14641, 512435, 12078825, 201957954, 2455383546, 21864605862, 141476861460, 648435615025, 1997181694277, 3709051717943, 3138428376721;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 2 + 55*x + 484*x^2 + 1331*x^3.","mathematica":"P[0, x_] := 2 + 55*x + 484*x^2 + 1331*x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + 11*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 + 55*x + 484*x^2 + 1331*x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 11*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-b1e393c17b0898c2","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-f27f365b1aa97cdc"],"text":"A013618 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A013618 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}