{"id":"n-009ca0264e18fd05","revision":1,"first_index":0,"fingerprint":"f6-d82797a2fb985b5a","name":"obreschkoff_combination of Triangle of 5-Eulerian numbers.","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","5"],["2","21","25"],["2","55","196","125"],["2","125","957","1651","625"],["2","267","3832","13132","12906","3125"],["2","553","13813","82398","154928","95561","15625"],["2","1127","46788","450863","1422218","1645581","679616","78125"],["2","2277","152389","2260565","11098065","21108583","16183943","4687071","390625"],["2","4579","483736","10682518","77715904","225254512","280603120","150120178","31559926","1953125"],["2","9185","1509213","48418536","504206724","2118867618","3971957010","3430266432","1330271634","208500181","9765625"],["2","18399","4653172","212926545","3093254028","18191582262","48786676176","62924661378","39269064930","11365733947","1356469836","48828125"],["2","36829","14229373","915708743","18194698713","145832232594","539700969570","987557178942","916458781572","426510756577","94274758769","8714990891","244140625"],["2","73691","43265112","3872764240","103620105934","1108867534119","5511615164400","13792088294472","18064060917390","12475266637321","4437949212184","763156497864","55414945346","1220703125"],["2","147417","131023525","16172580706","575449594560","8088348933791","52867030275471","175981467161340","313361602222938","304514786728135","160679489500227","44568543617282","6053823611780","349335375201","6103515625"],["2","294871","395674948","66885175683","3132878898566","57073755169211","482326783286128","2090106136874679","4919167854920814","6474939919635253","4800562336716212","1976592907751801","434493565556298","47212289730553","2186344282456","30517578125"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["7","48","378","3360","33264","362880","4324320","55883520","778377600","11623772160","185253868800","3138418483200","56317176115200","1067062284288000","21287892571545600"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","5"],["2","21","25"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","5"],["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":162801,"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-ab934b884ec7d157"]}],"region_exact":{},"incoming_edges":[{"id":"e-ab934b884ec7d157","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-009ca0264e18fd05","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":"6d42465d578107465443d683ceed1acb75402ecca0157c07955aa8c169101acc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-35f03041aed96954","role":"sequence","name":"Triangle of 5-Eulerian numbers.","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-35f03041aed96954","role":"sequence","name":"Triangle of 5-Eulerian numbers.","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,    5;\n  1: 2,   21,      25;\n  2: 2,   55,     196,      125;\n  3: 2,  125,     957,     1651,       625;\n  4: 2,  267,    3832,    13132,     12906,       3125;\n  5: 2,  553,   13813,    82398,    154928,      95561,      15625;\n  6: 2, 1127,   46788,   450863,   1422218,    1645581,     679616,      78125;\n  7: 2, 2277,  152389,  2260565,  11098065,   21108583,   16183943,    4687071,     390625;\n  8: 2, 4579,  483736, 10682518,  77715904,  225254512,  280603120,  150120178,   31559926,   1953125;\n  9: 2, 9185, 1509213, 48418536, 504206724, 2118867618, 3971957010, 3430266432, 1330271634, 208500181, 9765625;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n+3)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+5)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 5*x, P(1,x) = 2 + 21*x + 25*x^2.","mathematica":"P[0, x_] := 2 + 5*x;\nP[1, x_] := 2 + 21*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (n + 3)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 5)*x*P[n - 1, x] + (x - 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 + 5*x; v[2] = 2 + 21*x + 25*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n + 3)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 5)*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-35f03041aed96954","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-ab934b884ec7d157"],"text":"A144699 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A144699 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}