{"id":"n-9ae8757b995d759f","revision":1,"first_index":0,"fingerprint":"f6-7a7f16cb69475378","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.","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":false,"rows":[["2","1"],["2","21","1"],["2","239","220","1"],["2","2437","8541","2219","1"],["2","24435","228544","222440","22218","1"],["2","244433","5226469","12552654","5028464","222217","1"],["2","2444431","110346876","542871667","535545482","105562845","2222216","1"],["2","24444429","2229034405","20231306749","39790442625","19806108143","2125915991","22222215","1"],["2","244444427","43896097216","688125415586","2398111634464","2378552498588","670548341848","41794625654","222222214","1"],["2","2444444425","851670290349","22057748614248","126399721424772","219373210989882","124709294341770","21431096369616","810120109122","2222222213","1"],["2","24444444423","16380179959564","679412940763461","6082469782195788","17021922718317366","16928949228752256","5979818236468266","658833514684194","15572504294867","22222222212","1"],["2","244444444421","313427863674301","20359448017078975","274456022701281225","1171170143275885986","1865344827855499938","1160323663829329518","269155617405576420","19716994682354345","297879803823857","222222222211","1"],["2","2444444444419","5979373854253920","598446347533918796","11818425364796705710","73855490387671765971","177397610526477672672","176703435841898058720","72969622746543717678","11569629195408208265","579042780715238752","5681738494874484","2222222222210","1"],["2","24444444444417","113872547675266261","17351754398140703474","491563043303462186496","4364046959402132090671","15137930090142059465991","22645830108300578836668","15035082144687833173242","4302895762020385078703","480591860455587897027","16778690991279718642","108193253624836196","22222222222209","1"],["2","244444444444415","2166442850274500284","498210097198089849255","19917799337733968513798","245374447723174063802827","1189783695502413307617640","2552822081876144555482467","2545314181857986036111790","1179112660317127606535069","241563115004457140910620","19454559231676521320245","481545278754485396874","2058274041094108505","222222222222208","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","24","462","13200","497640","23274240","1298993520","84229474560","6221495280000","515637528806400","47383405772102400","4781138492101171200","525503368970061081600","62492292526169425920000","7994326521410223810816000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","1"],["2","21","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-17"],["0","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","9","-9"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","9","-9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162888,"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-c1abd10180bf6e94"]}],"region_exact":{},"incoming_edges":[{"id":"e-c1abd10180bf6e94","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-9ae8757b995d759f","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":"a46176d088b9582210b2bafffa06bc15b644d4ebb5e9b8cd322e966829e83c37","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-dcd2d11144e6ea21","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.","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-dcd2d11144e6ea21","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.","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":[],"oeis":[],"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,          1;\n  1: 2,         21,            1;\n  2: 2,        239,          220,              1;\n  3: 2,       2437,         8541,           2219,               1;\n  4: 2,      24435,       228544,         222440,           22218,               1;\n  5: 2,     244433,      5226469,       12552654,         5028464,          222217,               1;\n  6: 2,    2444431,    110346876,      542871667,       535545482,       105562845,         2222216,              1;\n  7: 2,   24444429,   2229034405,    20231306749,     39790442625,     19806108143,      2125915991,       22222215,            1;\n  8: 2,  244444427,  43896097216,   688125415586,   2398111634464,   2378552498588,    670548341848,    41794625654,    222222214,          1;\n  9: 2, 2444444425, 851670290349, 22057748614248, 126399721424772, 219373210989882, 124709294341770, 21431096369616, 810120109122, 2222222213, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(9*n-17)*x)*P(n-2,x) + (9*x-9*x^2)*d/dx P(n-2,x) + (9*n+1)*x*P(n-1,x) + (9*x-9*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 21*x + x^2.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 2 + 21*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (9*n - 17)*x)*P[n - 2, x] + (9*x - 9*x^2)*D[P[n - 2, x], x] + (9*n + 1)*x*P[n - 1, x] + (9*x - 9*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; v[2] = 2 + 21*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (9*n - 17)*x)*v[i - 2] + (9*x - 9*x^2)*deriv(v[i - 2], x) + (9*n + 1)*x*v[i - 1] + (9*x - 9*x^2)*deriv(v[i - 1], x)); 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-dcd2d11144e6ea21","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-c1abd10180bf6e94"],"text":"A257608 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257608 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}