{"id":"n-e92c310aea8cdd6a","revision":1,"first_index":0,"fingerprint":"f6-98515379e46553d3","name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":true,"rows":[["-4212","864","-53","1"],["153090","-35559","2781","-89","1"],["-6928416","1761264","-161190","6795","-134","1"],["375551640","-102364722","10491039","-528930","14040","-188","1"],["-23723526240","6840708552","-764776404","43875594","-1414665","25893","-251","1"],["1711537626960","-517192003908","62000505216","-3928641507","145859049","-3280662","43974","-323","1"],["-138851503257600","43666593782400","-5546211809616","380620169964","-15756084036","411827409","-6844824","70146","-404","1"],["12512256087300480","-4073562175547664","543390640077996","-39837784582620","1799993324205","-52850311542","1028259603","-13160880","106515","-494","1"],["-1239978636966182400","416192628042434880","-57919716432985968","4490792652763296","-218180254026240","7035910374240","-154710013689","2331820953","-23709510","155430","-593","1"],["134031568381456838400","-46225842241704994080","6676462999455338088","-543287324331671436","28070603157862386","-978537963455415","23753908190529","-406665758763","4893410313","-40500405","219483","-701","1"],["-15692967183952254873600","5546237904089478904320","-827898231622526842752","70281859081536015264","-3829527859158071928","142623347464525356","-3759146928208386","71354907436527","-979409113254","9633361023","-66185262","301509","-818","1"],["1978531422976739030400000","-714938758472830479523200","109922029473967506426240","-9688342840950324928104","553056984729230340996","-21808940283941744934","616491047298957831","-12753546362510382","194804710747218","-2193578509122","17974684728","-104181714","404586","-944","1"],["-267244196364314094314188800","98545599773319262958876160","-15561924435928486233985536","1418485780099049456019744","-84385756145522620872720","3498556403832800113440","-105069279178535103432","2338865681502087738","-39061041250444785","491024875634865","-4620758595453","32041936017","-158808195","532035","-1079","1"],["38501113513530474557726976000","-14464295348505053843900908800","2340459528510757030597549440","-219911709322856948909481072","13575046898282996134670544","-588375451240772747118840","18634115072476779679440","-441981365107362559479","7965417722899921713","-109788452614603620","1156576483190700","-9235737737226","54913964742","-235429740","687420","-1223","1"],["-5893093514574510679157538816000","2252431697017546351924704460800","-372695657340537971651376599040","35999805259481426999860983552","-2297658368798693750374907904","103628175518699048554784784","-3440346263309261778675840","86278429152829296891144","-1661043588183638369568","24767490026087727633","-286786403739950400","2569933674765756","-17639007265152","90939511782","-340614720","874548","-1376","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["-3400","120224","-5321680","283162880","-17605107520","1252560108800","-100365852146560","8943994925465600","-877426229449369600","93966017232165478400","-10908036245677322752000","1364358203111500960563200","-182923025215554060236800000","26170370855650847054028800000","-3979555055535695355194675200000"],"nonn":false,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["-4212","864","-53","1"]],"recurrence":{"denominator":{"coeffs":[["17","-1"],["9","0"]]},"terms":[{"coeff":{"coeffs":[["-468","44","-1"],["-396","18","0"],["-81","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162916,"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-07f8adddfb9eb09c"]}],"region_exact":{},"incoming_edges":[{"id":"e-07f8adddfb9eb09c","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-e92c310aea8cdd6a","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":"140c343d32e525b06d380d567b311998ad47f1632feaba9bb1fadea6b595ad3f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-59f3eb88935e03d3","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= 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-59f3eb88935e03d3","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= 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":[],"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:                 -4212,                   864,                 -53,                   1;\n  1:                153090,                -35559,                2781,                 -89,                 1;\n  2:              -6928416,               1761264,             -161190,                6795,              -134,                1;\n  3:             375551640,            -102364722,            10491039,             -528930,             14040,             -188,              1;\n  4:          -23723526240,            6840708552,          -764776404,            43875594,          -1414665,            25893,           -251,             1;\n  5:         1711537626960,         -517192003908,         62000505216,         -3928641507,         145859049,         -3280662,          43974,          -323,          1;\n  6:      -138851503257600,        43666593782400,      -5546211809616,        380620169964,      -15756084036,        411827409,       -6844824,         70146,       -404,         1;\n  7:     12512256087300480,     -4073562175547664,     543390640077996,     -39837784582620,     1799993324205,     -52850311542,     1028259603,     -13160880,     106515,      -494,      1;\n  8:  -1239978636966182400,    416192628042434880,  -57919716432985968,    4490792652763296,  -218180254026240,    7035910374240,  -154710013689,    2331820953,  -23709510,    155430,   -593,    1;\n  9: 134031568381456838400, -46225842241704994080, 6676462999455338088, -543287324331671436, 28070603157862386, -978537963455415, 23753908190529, -406665758763, 4893410313, -40500405, 219483, -701, 1;\n","recurrence_sentence":"The row polynomials satisfy (9*n+17-x)*P(n,x) = -(81*n^2+396*n+468-(18*n+44)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -4212 + 864*x - 53*x^2 + x^3.","mathematica":"P[0, x_] := -4212 + 864*x - 53*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(81*n^2 + 396*n + 468 - (18*n + 44)*x + x^2)*P[n - 1, x])/(9*n + 17 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -4212 + 864*x - 53*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (-(81*n^2 + 396*n + 468 - (18*n + 44)*x + x^2)*v[i - 1])/(9*n + 17 - 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-4f6e747b794bcc2e","steps":["x \u21a6 (\u22122/9)x+1 (shift +2)","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-2bc76bcaa23d62fb","e-07f8adddfb9eb09c"],"text":"A028338 \u2192 x \u21a6 (\u22122/9)x+1 (shift +2) \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 (\u22122/9)x+1 (shift +2) \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}