{"id":"n-9eedfe3e5b2b8d6d","revision":1,"first_index":0,"fingerprint":"f6-f449864e9c351c80","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.","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":[["5","4"],["20","60","16"],["80","608","568","64"],["320","5248","11592","4760","256"],["1280","41664","182488","176144","37224","1024"],["5120","315200","2477416","4629096","2342936","279256","4096"],["20480","2315968","30652056","99302560","97150880","28624320","2041832","16384"],["81920","16711488","356637160","1866404440","3129882096","1799905616","330328808","14690136","65536"],["327680","119225024","3978045592","32024236976","85255263160","83991785504","30580968520","3664428272","104616808","262144"],["1310720","844536640","43074735336","514781347272","2063302745448","3206588813064","2011335196920","488178424344","39529532088","740247512","1048576"],["5242880","5955534528","456614335384","7887300840384","45791619576624","106071187170144","104927901102528","44295065726976","7442197151376","417929339936","5216597736","4194304"],["20971520","41879582528","4766411571688","116571680379448","952056750747264","3154199798089920","4632606836237520","3095063516564016","915648985860672","109586097563456","4354437766280","36668227672","16777216"],["83886080","293983355584","49198013622936","1676238270306512","18824421787207432","86455953573460416","180463504893724272","178985097855576672","84312946807643664","18025344481726208","1571418584314744","44885455439184","257336099432","67108864"],["335544320","2061440258880","503629478352616","23598376351381928","357855343061113656","2223495454658107960","6381810048780305040","8972626088679560208","6289280904498188784","2158006929637671728","341506236849693960","22076155780690120","459038006269016","1804188045528","268435456"],["1342177280","14445315524288","5124164756691352","326789525327852256","6594124205892656960","54359746775861363584","209077966065786196984","402478532048209336896","399887716008310081728","205012180615946116160","52559595189284805512","6276216345394344928","305216999025514752","4667188190681216","12641463023080","1073741824"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["9","96","1320","22176","439824","10053120","260124480","7514707200","239719159680","8368377937920","317300996812800","12984933100339200","570409561193472000","26771222072013619200","1336887902221180108800"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["5","4"],["20","60","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","-2"],["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3","-3"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"],["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":162858,"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-84efbcedb981a87c"]}],"region_exact":{},"incoming_edges":[{"id":"e-84efbcedb981a87c","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-9eedfe3e5b2b8d6d","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":"3feb17a0842f31317270083460d8908da4bebfeb5ff29943af7f3999b17d7a77","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-e224ad3e3147ef52","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.","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-e224ad3e3147ef52","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.","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:       5,         4;\n  1:      20,        60,          16;\n  2:      80,       608,         568,           64;\n  3:     320,      5248,       11592,         4760,           256;\n  4:    1280,     41664,      182488,       176144,         37224,          1024;\n  5:    5120,    315200,     2477416,      4629096,       2342936,        279256,          4096;\n  6:   20480,   2315968,    30652056,     99302560,      97150880,      28624320,       2041832,        16384;\n  7:   81920,  16711488,   356637160,   1866404440,    3129882096,    1799905616,     330328808,     14690136,       65536;\n  8:  327680, 119225024,  3978045592,  32024236976,   85255263160,   83991785504,   30580968520,   3664428272,   104616808,    262144;\n  9: 1310720, 844536640, 43074735336, 514781347272, 2063302745448, 3206588813064, 2011335196920, 488178424344, 39529532088, 740247512, 1048576;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+(3*n-2)*x)*P(n-2,x) + (3*x-3*x^2)*d/dx P(n-2,x) + (3+(3*n+4)*x)*P(n-1,x) + (3*x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 5 + 4*x, P(1,x) = 20 + 60*x + 16*x^2.","mathematica":"P[0, x_] := 5 + 4*x;\nP[1, x_] := 20 + 60*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(4 + (3*n - 2)*x)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (3 + (3*n + 4)*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] = 5 + 4*x; v[2] = 20 + 60*x + 16*x^2; for(n = 2, N, my(i = n + 1); v[i] = (4 + (3*n - 2)*x)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (3 + (3*n + 4)*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-e224ad3e3147ef52","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-84efbcedb981a87c"],"text":"A257622 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257622 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}