{"id":"n-14ac68f63102aec9","revision":1,"first_index":1,"fingerprint":"f6-bfd323cc05a9e10a","name":"obreschkoff_combination of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4, read by rows.","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","27","20","1"],["2","61","133","43","1"],["2","131","624","552","90","1"],["2","273","2485","4590","2064","185","1"],["2","559","9004","30275","28170","7213","376","1"],["2","1133","30741","173149","288145","153983","24087","759","1"],["2","2283","100912","900754","2453424","2338428","777512","77958","1526","1"],["2","4585","322525","4386600","18425444","28489162","16987770","3710000","246850","3061","1"],["2","9191","1011900","20371925","126466220","297721606","287657888","113865050","16984450","769635","6132","1"],["2","18405","3133741","91361375","812045625","2781825426","4071358770","2620633758","718552500","75388825","2373297","12275","1"],["2","36835","9615952","399150076","4957102750","23889130931","50482690880","49193157536","22080542798","4329911625","326894992","7261060","24562","1"],["2","73697","29314437","1709624562","29101542336","192203999711","565870613031","793606852476","540566586426","175080559663","25170780787","1392345522","22090212","49137","1"],["2","147423","88950508","7212337143","165698678598","1469085674587","5861755558904","11395007581059","11167271341614","5513392972189","1323106657260","142237522373","5849306058","66933993","98288","1"],["2","294877","269013733","30072590033","920868679113","10778051937737","57021406062497","149258428380509","202313586207609","143952912597799","52977098764895","9620643959075","785843493883","24312234483","202227163","196591","1"],["2","589787","811660944","124258681270","5019532415620","76498752029254","527188205286024","1815960326029638","3302758476188960","3249703318361860","1729028819494336","484989158458866","67798976749292","4263761511770","100215460520","609728658","393198","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["50","240","1400","9600","75600","672000","6652800","72576000","864864000","11176704000","155675520000","2324754432000","37050773760000","627683696640000","11263435223040000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","27","20","1"],["2","61","133","43","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162798,"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-8df0a750c7bea2d2"]}],"region_exact":{},"incoming_edges":[{"id":"e-8df0a750c7bea2d2","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-14ac68f63102aec9","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":"ed1e5d06d039c77a002efc0700a3f003d4a16e992ed6f3a57a2e2823c53a7642","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-2255c682502f6ca3","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4, read by rows.","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-2255c682502f6ca3","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x)^4, read by rows.","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,    27,      20,        1;\n   2: 2,    61,     133,       43,         1;\n   3: 2,   131,     624,      552,        90,          1;\n   4: 2,   273,    2485,     4590,      2064,        185,          1;\n   5: 2,   559,    9004,    30275,     28170,       7213,        376,          1;\n   6: 2,  1133,   30741,   173149,    288145,     153983,      24087,        759,         1;\n   7: 2,  2283,  100912,   900754,   2453424,    2338428,     777512,      77958,      1526,        1;\n   8: 2,  4585,  322525,  4386600,  18425444,   28489162,   16987770,    3710000,    246850,     3061,       1;\n   9: 2,  9191, 1011900, 20371925, 126466220,  297721606,  287657888,  113865050,  16984450,   769635,    6132,     1;\n  10: 2, 18405, 3133741, 91361375, 812045625, 2781825426, 4071358770, 2620633758, 718552500, 75388825, 2373297, 12275, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+2)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 27*x + 20*x^2 + x^3, P(2,x) = 2 + 61*x + 133*x^2 + 43*x^3 + x^4.","mathematica":"P[1, x_] := 2 + 27*x + 20*x^2 + x^3;\nP[2, x_] := 2 + 61*x + 133*x^2 + 43*x^3 + x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 2)*x*P[n - 1, x] + (x - x^2)*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 + 27*x + 20*x^2 + x^3; v[2] = 2 + 61*x + 133*x^2 + 43*x^3 + x^4; for(n = 3, N, my(i = n); v[i] = (1 + n*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 2)*x*v[i - 1] + (x - 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2255c682502f6ca3","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-8df0a750c7bea2d2"],"text":"A166340 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A166340 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}