{"id":"n-b95e01c853e1cae8","revision":1,"first_index":1,"fingerprint":"f6-18f988f21afcd5d4","name":"obreschkoff_combination of Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2","122","31","1"],["2","1222","711","59","1"],["2","12222","14731","2363","96","1"],["2","122222","292111","80895","5915","142","1"],["2","1222222","5672331","2557171","299750","12447","197","1"],["2","12222222","108996511","77273119","13647921","872312","23282","261","1"],["2","122222222","2083155931","2272643843","582246196","53774273","2152822","39986","334","1"],["2","1222222222","39702184911","65717183535","23815753095","3055862754","172179483","4711926","64368","416","1"],["2","12222222222","755563735531","1879783323891","946900048050","164385439779","12525734319","473742747","9410790","98480","507","1"],["2","122222222222","14367933197311","53389496804479","36915085101741","8508630277884","853300827324","42845270127","1160730417","17486150","144617","607","1"],["2","1222222222222","273112952971131","1509273843722723","1419247645568896","428312077884405","55440175780704","3595398115452","127578590568","2594594717","30646297","205317","716","1"],["2","12222222222222","5190368328673711","42532780577207375","54021436729771875","21121603228251526","3477521745823125","285545655169632","12908635226916","340335357362","5383407744","51177997","283361","834","1"],["2","122222222222222","98629220467022731","1196108224490480211","2041325939578766750","1025615185229342071","212385299248523401","21752443676679573","1227876026734500","40816134530600","830225462066","10501207444","82064346","381773","961","1"],["2","1222222222222222","1874077411095654111","33589659506200468639","76725167988904849961","49219624460128502016","12706806643898129126","1604541694556016073","111387393628298073","4574799058243700","116366651578606","1880346206466","19446221158","127113560","503820","1097","1"],["2","12222222222222222","35608693033039650331","942384543584708776003","2872420875095679917196","2340827893154815942697","748093989874525603946","115397475095483157798","9735821429421775402","486520916404281473","15164164351896846","304401272225206","3999984312688","34445621238","191098700","653012","1242","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["156","1995","29415","501288","9764121","213035631","5116235610","133690162713","3771863756319","114200692062882","3690334520303151","126654732892573683","4597208067883418652","175837180386264008631","7064983507006547136003"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","122","31","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162885,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-a20ccc97cc2b4079"]}],"region_exact":{},"incoming_edges":[{"id":"e-a20ccc97cc2b4079","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-b95e01c853e1cae8","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":"6021929017d85b19efdd0df78b1c44539accbbba96d53fcf20b7e1dcd2fd4e24","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-2b620f031a59b07c","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-2b620f031a59b07c","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 2,          122,             31,              1;\n   2: 2,         1222,            711,             59,              1;\n   3: 2,        12222,          14731,           2363,             96,             1;\n   4: 2,       122222,         292111,          80895,           5915,           142,            1;\n   5: 2,      1222222,        5672331,        2557171,         299750,         12447,          197,           1;\n   6: 2,     12222222,      108996511,       77273119,       13647921,        872312,        23282,         261,          1;\n   7: 2,    122222222,     2083155931,     2272643843,      582246196,      53774273,      2152822,       39986,        334,        1;\n   8: 2,   1222222222,    39702184911,    65717183535,    23815753095,    3055862754,    172179483,     4711926,      64368,      416,      1;\n   9: 2,  12222222222,   755563735531,  1879783323891,   946900048050,  164385439779,  12525734319,   473742747,    9410790,    98480,    507,   1;\n  10: 2, 122222222222, 14367933197311, 53389496804479, 36915085101741, 8508630277884, 853300827324, 42845270127, 1160730417, 17486150, 144617, 607, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 + 122*x + 31*x^2 + x^3.","mathematica":"P[1, x_] := 2 + 122*x + 31*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + 122*x + 31*x^2 + x^3; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 9*x*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2b620f031a59b07c","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a20ccc97cc2b4079"],"text":"A166979 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A166979 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}