{"id":"n-88fcd1b546ee4111","revision":1,"first_index":0,"fingerprint":"f6-84b877d940ba685c","name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*4^(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":[["-352","164","-23","1"],["5760","-3024","536","-39","1"],["-116736","66944","-13840","1320","-59","1"],["2826240","-1736192","401344","-45680","2740","-83","1"],["-79626240","51720192","-13031424","1685824","-122640","5068","-111","1"],["2559836160","-1741897728","470388736","-67165952","5621504","-285152","8624","-143","1"],["-92484403200","65482260480","-18729680896","2895294464","-270041856","15907584","-596064","13776","-179","1"],["3709945774080","-2718956519424","816604839936","-134817054720","13719956480","-907506432","39785088","-1147680","20940","-219","1"],["-163618121318400","123613143367680","-38726423543808","6760412020736","-739598295040","53715031040","-2660469504","90338688","-2069760","30580","-263","1"],["7868890113638400","-6108193939783680","1985824746897408","-363777325531136","42317113163776","-3321614602240","181579126784","-7001372928","189771648","-3538480","43208","-311","1"],["-409851978684825600","326000572492677120","-109529329746051072","20929838447591424","-2567281586208768","215259627421696","-12774532632576","546016520192","-16877862144","373895808","-5786352","59384","-363","1"],["22983856059580416000","-18690822411544166400","6467744586226728960","-1283082984886370304","164869890104098816","-14635325735174144","931370574790656","-43379829256192","1491944250368","-37830304512","698105408","-9113104","79716","-419","1"],["-1380702916742794444800","1145762141415470530560","-407201716452087300096","83537925735274512384","-11185717800305950720","1043863977737584640","-70569395473088512","3536372767735808","-132964763176960","3763272381440","-79739948288","1245130432","-13897520","104860","-479","1"],["88458593648945135616000","-74785572197642325196800","27232989191560904048640","-5758846497505067139072","800093785924445405184","-78052428221975101440","5564082605564887040","-297072984362778624","12052153666961408","-373967026667520","8869444812800","-159464987136","2134892032","-20610240","135520","-543","1"],["-6020800786772646690816000","5178536643884594547916800","-1928283961286803996016640","419173920258841978601472","-60210636732538168541184","6111893776776166375424","-456696100299829411840","25779366765059375104","-1117157328824762368","37497117400956928","-977411049472000","19718081150976","-304693555712","3536809472","-29827520","172448","-611","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["-210","3234","-62370","1448370","-39369330","1226506050","-43091245890","1685628148050","-72659444908050","3421913857812450","-174815164475201250","9628819259294084850","-568813582169409827250","35874484199581054277250","-2405904924223516511045250"],"nonn":false,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["-352","164","-23","1"]],"recurrence":{"denominator":{"coeffs":[["7","-1"],["4","0"]]},"terms":[{"coeff":{"coeffs":[["-88","19","-1"],["-76","8","0"],["-16","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162906,"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-f961be0564a72cf6"]}],"region_exact":{},"incoming_edges":[{"id":"e-f961be0564a72cf6","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-88fcd1b546ee4111","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":"6c3641eac2044a63a09591fd12f09e128bd9bcf34cfe71dac491a58159c6f5b2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-552ecb7379159c73","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*4^(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-552ecb7379159c73","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*4^(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:             -352,               164,              -23,                1;\n  1:             5760,             -3024,              536,              -39,              1;\n  2:          -116736,             66944,           -13840,             1320,            -59,              1;\n  3:          2826240,          -1736192,           401344,           -45680,           2740,            -83,            1;\n  4:        -79626240,          51720192,        -13031424,          1685824,        -122640,           5068,         -111,           1;\n  5:       2559836160,       -1741897728,        470388736,        -67165952,        5621504,        -285152,         8624,        -143,         1;\n  6:     -92484403200,       65482260480,     -18729680896,       2895294464,     -270041856,       15907584,      -596064,       13776,      -179,        1;\n  7:    3709945774080,    -2718956519424,     816604839936,    -134817054720,    13719956480,     -907506432,     39785088,    -1147680,     20940,     -219,     1;\n  8: -163618121318400,   123613143367680,  -38726423543808,    6760412020736,  -739598295040,    53715031040,  -2660469504,    90338688,  -2069760,    30580,  -263,    1;\n  9: 7868890113638400, -6108193939783680, 1985824746897408, -363777325531136, 42317113163776, -3321614602240, 181579126784, -7001372928, 189771648, -3538480, 43208, -311, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n+7-x)*P(n,x) = -(16*n^2+76*n+88-(8*n+19)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -352 + 164*x - 23*x^2 + x^3.","mathematica":"P[0, x_] := -352 + 164*x - 23*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(16*n^2 + 76*n + 88 - (8*n + 19)*x + x^2)*P[n - 1, x])/(4*n + 7 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -352 + 164*x - 23*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (-(16*n^2 + 76*n + 88 - (8*n + 19)*x + x^2)*v[i - 1])/(4*n + 7 - 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 (\u22121/2)x+1 (shift +2)","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-41f01d4be10aeae3","e-f961be0564a72cf6"],"text":"A028338 \u2192 x \u21a6 (\u22121/2)x+1 (shift +2) \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 (\u22121/2)x+1 (shift +2) \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}