{"id":"n-09ab1250d4ac8bc2","revision":1,"first_index":0,"fingerprint":"f6-a521dd3b171d6b2b","name":"obreschkoff_combination of Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.","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","1"],["2","15","1"],["2","125","112","1"],["2","907","2997","797","1"],["2","6393","55528","53438","5598","1"],["2","44807","876913","2053038","829868","39211","1"],["2","313717","12755304","60614695","59438570","11970243","274508","1"],["2","2196099","177191221","1542148003","2989329249","1495464737","165534695","1921593","1"],["2","15372785","2396271088","35785301036","122188133524","120740952278","34467027520","2232932492","13451194","1"],["2","107609599","31893261057","781968815340","4369858274208","7513217370162","4284902622966","749898058908","29675701350","94158407","1"],["2","753267309","420450214528","16406791217853","142665305310828","393878950109274","390735591013320","139162371503202","15688786420002","390882122777","659108904","1"],["2","5272871291","5511237144277","334675821437305","4366079400157449","18319351976619294","28986374261826450","18071281690916802","4243323819872100","319407485231831","5121108288809","4613762389","1"],["2","36910099177","71995410598680","6692681345323466","127465200437968930","781007313656181951","1857909072741262176","1847242050456055020","767302111365422598","123556552439534135","6378785097235912","66879575181162","32296336790","1"],["2","258370694391","938607092448433","131953915376177978","3592969472830349916","31196446759255718101","106904396858660846463","159149898961854626844","105838162121860973394","30556609848328264433","3476105601736139247","125647125689540746","871764427366448","226074357603","1"],["2","1808594860901","12222109708665784","2575241460127604547","98626907163074336390","1185521562165125005957","5666688808849497492628","12070785246333312870507","12018539744230119090126","5592047093486892237335","1158557921789627091440","95282755476714375121","2450532941530512234","11350603651993727","1582520503300","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","18","240","4704","120960","3843840","145367040","6373785600","317839441920","17761615872000","1099350540288000","74651136688128000","5517692711731200000","440974001521557504000","37891099390000496640000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","1"],["2","15","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-11"],["0","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","6","-6"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162853,"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-5de672ddb1a17fcf"]}],"region_exact":{},"incoming_edges":[{"id":"e-5de672ddb1a17fcf","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-09ab1250d4ac8bc2","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":"b461565f3564b292573b8f869748c88d220e7214cfe2d95fdff0ca67bf49613b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-3bc26d3108ee6bfd","role":"sequence","name":"Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.","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-3bc26d3108ee6bfd","role":"sequence","name":"Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.","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,         1;\n  1: 2,        15,           1;\n  2: 2,       125,         112,            1;\n  3: 2,       907,        2997,          797,             1;\n  4: 2,      6393,       55528,        53438,          5598,             1;\n  5: 2,     44807,      876913,      2053038,        829868,         39211,             1;\n  6: 2,    313717,    12755304,     60614695,      59438570,      11970243,        274508,            1;\n  7: 2,   2196099,   177191221,   1542148003,    2989329249,    1495464737,     165534695,      1921593,           1;\n  8: 2,  15372785,  2396271088,  35785301036,  122188133524,  120740952278,   34467027520,   2232932492,    13451194,        1;\n  9: 2, 107609599, 31893261057, 781968815340, 4369858274208, 7513217370162, 4284902622966, 749898058908, 29675701350, 94158407, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(6*n-11)*x)*P(n-2,x) + (6*x-6*x^2)*d/dx P(n-2,x) + (6*n+1)*x*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 15*x + x^2.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 2 + 15*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (6*n - 11)*x)*P[n - 2, x] + (6*x - 6*x^2)*D[P[n - 2, x], x] + (6*n + 1)*x*P[n - 1, x] + (6*x - 6*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 + x; v[2] = 2 + 15*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (6*n - 11)*x)*v[i - 2] + (6*x - 6*x^2)*deriv(v[i - 2], x) + (6*n + 1)*x*v[i - 1] + (6*x - 6*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-3bc26d3108ee6bfd","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-5de672ddb1a17fcf"],"text":"A142461 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A142461 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}