{"id":"n-db81f1fdf59f6128","revision":1,"first_index":0,"fingerprint":"f6-8ddf4efa9ad6adc7","name":"obreschkoff_combination of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","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":[["4","3"],["12","39","9"],["36","333","306","27"],["108","2403","5643","2133","81"],["324","15957","78732","75492","13932","243"],["972","101331","941625","1834650","882090","87723","729"],["2916","627669","10257030","35920017","35026992","9476271","540918","2187"],["8748","3833811","105231879","613062027","1058921343","588292065","96383277","3295809","6561"],["26244","23232501","1037477808","9538447410","26816460372","26370601056","9055387224","944928342","19945440","19683"],["78732","140162643","9954696933","139073307048","600817591368","954292817934","583985437722","131055397632","9033001092","120243447","59049"],["236196","843514965","93737335050","1934831597607","12323144607264","29717065048194","29363589821628","11878232476590","1813730896908","84844496601","723350250","177147"],["708588","5069415699","871111338003","26003060036145","236697280955613","828937563761070","1237908248905806","811897118546706","226753880076630","24283066474287","787110356367","4346301645","531441"],["2125764","30443597685","8020331133588","340529221769328","4325136861914460","21276458377359777","45804209065503480","45395238380358744","20701258539768684","4124386041873975","317117266633044","7238290192920","26098004628","1594323"],["6377292","182749273875","73357242138513","4372723420845606","76041922220500374","512055509163183981","1533952319564787975","2185159379645760084","1509833539561789008","495679387485329757","72258065400395727","4063626485346150","66149395543278","156653395011","4782969"],["19131876","1096777838421","667812356329806","55311911661344013","1297097883819243792","11714082035114806425","47513235057623486658","93726752471605940193","93075545411524968084","46515457027224881631","11294444569849977042","1229212541839693671","51321642418136376","601845710060403","940130820702","14348907"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["7","60","702","10368","184680","3849120","91854000","2469035520","73806526080","2428472793600","87210743558400","3394147857408000","142299648921830400","6393556318999449600","306473732247691008000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","3"],["12","39","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","-3"],["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162847,"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-9a566c54793e3e13"]}],"region_exact":{},"incoming_edges":[{"id":"e-9a566c54793e3e13","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-db81f1fdf59f6128","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":"7e622bd23c1ad9f167cf7fdd1064858e7dd303ddb118ac0ebb94cd2b38c87b14","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-dfedc6a36c97eba4","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","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-dfedc6a36c97eba4","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","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:     4,         3;\n  1:    12,        39,          9;\n  2:    36,       333,        306,           27;\n  3:   108,      2403,       5643,         2133,           81;\n  4:   324,     15957,      78732,        75492,        13932,          243;\n  5:   972,    101331,     941625,      1834650,       882090,        87723,          729;\n  6:  2916,    627669,   10257030,     35920017,     35026992,      9476271,       540918,         2187;\n  7:  8748,   3833811,  105231879,    613062027,   1058921343,    588292065,     96383277,      3295809,       6561;\n  8: 26244,  23232501, 1037477808,   9538447410,  26816460372,  26370601056,   9055387224,    944928342,   19945440,     19683;\n  9: 78732, 140162643, 9954696933, 139073307048, 600817591368, 954292817934, 583985437722, 131055397632, 9033001092, 120243447, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(3*n-3)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (2+(3*n+3)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4 + 3*x, P(1,x) = 12 + 39*x + 9*x^2.","mathematica":"P[0, x_] := 4 + 3*x;\nP[1, x_] := 12 + 39*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + (3*n - 3)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (2 + (3*n + 3)*x)*P[n - 1, x] + (3*x - 3*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] = 4 + 3*x; v[2] = 12 + 39*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (3 + (3*n - 3)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (2 + (3*n + 3)*x)*v[i - 1] + (3*x - 3*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-dfedc6a36c97eba4","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-9a566c54793e3e13"],"text":"A257620 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257620 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}