{"id":"n-f3255eb099254811","revision":1,"first_index":0,"fingerprint":"f6-a8b6c362e98a3f2e","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) = 9*x + 2.","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":[["3","2"],["6","46","4"],["12","608","568","8"],["24","7000","23124","6444","16"],["48","77840","660404","644280","71420","32"],["96","858352","16108076","37632796","15541636","786980","64"],["192","9446960","361861428","1697717376","1676192656","347178144","8660076","128"],["384","103928464","7759175788","66102058932","127875626648","64787727704","7421444604","95268580","256"],["768","1143240368","161860925812","2348356032392","7951124080612","7889350512896","2291327480476","154543409672","1047972172","512"],["1536","12575705488","3320960887404","78556570055820","433070680848636","743366371756860","427529686368132","76427103501156","3167560718100","11527734084","1024"],["3072","138332897072","67453569556340","2522501604015696","21542307693613512","59119753417508544","58809457726600320","21193334577301200","2449395461401944","64298584543728","126805164524","2048"],["6144","1521662168848","1361694268055276","78722694174915748","1004586357156235536","4175051198762729616","6597415950906872136","4137784048729742808","985915524182950032","76340752283070144","1297534016408620","1394857007396","4096"],["12288","16738284514736","27386431993687668","2407655420222350680","44686249127917055612","270376743214112250240","640308163262767912008","637885798510623769488","267243545522538742968","43779056297909021328","2332676362207335812","26090419639447896","15343427513484","8192"],["24576","184121131087504","549557297457326188","72576360152909418108","1918895277334268035924","16407814385023282728628","55828086137678039376312","83023756229619891800664","55460577082381527857064","16185257088628661041272","1877523876456581365276","70271070222695769964","523483616102393028","168777703586308","16384"],["49152","2025332445034544","11012918272898352244","2164930945931637022880","80227250503551071777792","947312316592352802698464","4485825076980856931511332","9522118606929753537178752","9494922936837811684754400","4446772314283498686382896","933045954781058409693116","78422302987247399830624","2095209433006398579104","10489618777927046720","1856554741472812","32768"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","56","1196","36608","1454024","70928000","4101056960","274145231360","20798753655680","1765463037578240","165768767772608000","17058478671000166400","1909073396651643622400","230845830901310251827200","29992928406694006571417600"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3","2"],["6","46","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-16"],["0","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","9","-9"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["0","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","9","-9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162892,"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-438bbc5de73b4ca9"]}],"region_exact":{},"incoming_edges":[{"id":"e-438bbc5de73b4ca9","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-f3255eb099254811","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":"0558497f14834567fbf27ce7c03b6cd254f06d76c3f87b4d0c6dd8582a469c28","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-6a6e9e081117dc37","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) = 9*x + 2.","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-6a6e9e081117dc37","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) = 9*x + 2.","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:    3,           2;\n  1:    6,          46,             4;\n  2:   12,         608,           568,              8;\n  3:   24,        7000,         23124,           6444,              16;\n  4:   48,       77840,        660404,         644280,           71420,              32;\n  5:   96,      858352,      16108076,       37632796,        15541636,          786980,              64;\n  6:  192,     9446960,     361861428,     1697717376,      1676192656,       347178144,         8660076,            128;\n  7:  384,   103928464,    7759175788,    66102058932,    127875626648,     64787727704,      7421444604,       95268580,           256;\n  8:  768,  1143240368,  161860925812,  2348356032392,   7951124080612,   7889350512896,   2291327480476,   154543409672,    1047972172,         512;\n  9: 1536, 12575705488, 3320960887404, 78556570055820, 433070680848636, 743366371756860, 427529686368132, 76427103501156, 3167560718100, 11527734084, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(9*n-16)*x)*P(n-2,x) + (9*x-9*x^2)*d/dx P(n-2,x) + (1+(9*n+2)*x)*P(n-1,x) + (9*x-9*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 46*x + 4*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 6 + 46*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (9*n - 16)*x)*P[n - 2, x] + (9*x - 9*x^2)*D[P[n - 2, x], x] + (1 + (9*n + 2)*x)*P[n - 1, x] + (9*x - 9*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] = 3 + 2*x; v[2] = 6 + 46*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (9*n - 16)*x)*v[i - 2] + (9*x - 9*x^2)*deriv(v[i - 2], x) + (1 + (9*n + 2)*x)*v[i - 1] + (9*x - 9*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-6a6e9e081117dc37","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-438bbc5de73b4ca9"],"text":"A257619 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257619 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}