{"id":"n-8ae9deae4758feb6","revision":1,"first_index":0,"fingerprint":"f6-48ae66378f7b3d5e","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) = 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","44","16"],["80","336","312","64"],["320","2224","3848","1992","256"],["1280","13616","38360","36736","11944","1024"],["5120","79344","337160","522120","314040","68680","4096"],["20480","446896","2725368","6317312","6132352","2490672","383336","16384"],["81920","2455664","20753512","68447144","99110000","65040160","18709736","2092808","65536"],["327680","13244976","151154136","684242112","1403074296","1372411440","639955464","134900064","11234088","262144"],["1310720","70419184","1063788104","6434059784","18004398344","24886091624","17316229016","5945258456","942133016","59512776","1048576"],["5242880","370183856","7287009464","57671052096","214188219408","402709013760","395827320480","203306050176","52789581744","6415295632","311981800","4194304"],["20971520","1928513904","48845380136","497497523240","2399745945600","5963756933280","7961787045072","5782117832208","2254110947520","451994950960","42800268360","1621774984","16777216"],["83886080","9973919536","321689056152","4159961939584","25613576749640","82273270131360","144840064000368","142842033888576","78890898244752","23856963325360","3756812368760","280817301696","8373116072","67108864"],["335544320","51278883824","2088099468808","33902580084232","262702442245336","1071233676597320","2430095257363152","3156920422449744","2369526493605936","1017956355102208","243005440859560","30467456771624","1817256086648","42989653832","268435456"],["1342177280","262367108016","13391981499000","270448452369536","2606443916192544","13295525956391184","38150301723482168","63750010325633280","63023185160546688","36852008906473552","12540272043534312","2397341055417216","242069095066720","11626004296176","219712998504","1073741824"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["9","80","792","8640","102960","1330560","18532800","276756480","4410806400","74724249600","1340885145600","25406244864000","506854585036800","10619810353152000","233153109116928000"],"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","44","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","2"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162824,"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-7b496eddb4982c99"]}],"region_exact":{},"incoming_edges":[{"id":"e-7b496eddb4982c99","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-8ae9deae4758feb6","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":"e3e65c1c96ccfa5dd807075f6f584c288cef835ce096993920253f6779da7c00","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c47b645304a2946e","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) = 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-c47b645304a2946e","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) = 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,       44,         16;\n  2:      80,      336,        312,         64;\n  3:     320,     2224,       3848,       1992,         256;\n  4:    1280,    13616,      38360,      36736,       11944,        1024;\n  5:    5120,    79344,     337160,     522120,      314040,       68680,        4096;\n  6:   20480,   446896,    2725368,    6317312,     6132352,     2490672,      383336,      16384;\n  7:   81920,  2455664,   20753512,   68447144,    99110000,    65040160,    18709736,    2092808,     65536;\n  8:  327680, 13244976,  151154136,  684242112,  1403074296,  1372411440,   639955464,  134900064,  11234088,   262144;\n  9: 1310720, 70419184, 1063788104, 6434059784, 18004398344, 24886091624, 17316229016, 5945258456, 942133016, 59512776, 1048576;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+(n+2)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (3+(n+4)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 5 + 4*x, P(1,x) = 20 + 44*x + 16*x^2.","mathematica":"P[0, x_] := 5 + 4*x;\nP[1, x_] := 20 + 44*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(4 + (n + 2)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (3 + (n + 4)*x)*P[n - 1, x] + (x - 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 + 44*x + 16*x^2; for(n = 2, N, my(i = n + 1); v[i] = (4 + (n + 2)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (3 + (n + 4)*x)*v[i - 1] + (x - 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-c47b645304a2946e","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-7b496eddb4982c99"],"text":"A257606 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257606 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}