{"id":"n-bb1f3df8492d87c4","revision":1,"first_index":0,"fingerprint":"f6-ead18c54fb42257e","name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1872","3192","1813","343"],["22464","51408","44100","16807","2401"],["269568","774144","889056","510384","146461","16807"],["3234816","11176704","16087680","12348000","5330220","1226911","117649"],["38817792","156764160","271289088","260789760","150398640","52034472","10000165","823543"],["465813504","2152894464","4352818176","5028500736","3630312000","1677204144","484243284","79883671","5764801"],["5589762048","29095428096","67304079360","90811736064","78763249152","45538633728","17551348416","4348307040","628363309","40353607"],["67077144576","388273471488","1011316948992","1560869388288","1580841142272","1097806348800","529386617088","175039123392","37978508988","4882786447","282475249"],["804925734912","5128821669888","14853717688320","25809651302400","29896179425280","24239564181504","14037283846656","5806175800320","1681015971600","324443000280","37569208117","1977326743"],["9659108818944","67180340183040","214146363949056","413691839447040","539421712220160","500148026155008","338124355430400","167935096530432","60815422261440","15660427804560","2721931499364","286712377735","13841287201"],["115909305827328","873777843929088","3040018748669952","6463326621007872","9368903422771200","9777728299401216","7558528448249856","4382091646377984","1905330742850304","613633089484800","142286172624288","22494069028368","2173082090557","96889010407"],["1390911669927936","11296699267940352","42596669891543040","98840050692784128","157670127420309504","182915063552212992","159146439474806784","105494798894284800","53538610438849536","20700912273769728","6002865697885056","1265932036710432","183535468285260","16374242758783","678223072849"],["16690940039135232","145296772904779776","590236933574098944","1484257297554210816","2583921883893202944","3298671654568722432","3190162718563172352","2379962663055065088","1380926917526188032","623181220357183488","216940774291008768","57211244325720576","11063949876396144","1481239191102216","122758376185669","4747561509943"],["200291280469622784","1860397855131303936","8099920613222645760","21942746105669222400","41396863689597911040","57671513042077089792","61372654204739125248","50890690986602987520","33230861651699712000","17144663066969518080","6965557833992389632","2205120351945708288","533246108796797760","95222519427999600","11841774851943540","916279371418999","33232930569601"],["2403495365635473408","23726813224863006720","110221832344590876672","320012397560589189120","650361587014859489280","981836202332110454784","1140172441751409131520","1040296871272409726976","755005176726617456640","438351988365532200960","203599335476695302144","75220349061295226880","21834795769181531136","4875392994713579520","808658934219319680","93887776420632768","6812750766768205","232630513987207"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["7220","137180","2606420","49521980","940917620","17877434780","339671260820","6453753955580","122621325156020","2329805177964380","44266298381323220","841059669245141180","15980133715657682420","303622540597495965980","5768828271352423353620"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1872","3192","1813","343"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["12","7"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162905,"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-946cc0d28b50f639"]}],"region_exact":{},"incoming_edges":[{"id":"e-946cc0d28b50f639","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-bb1f3df8492d87c4","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":"e4032c8da9e638123f584edf8803ec50e2c6066d89d51386af3f7a15198fb023","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-5cf6acfdac607d6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","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-5cf6acfdac607d6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","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":[{"id":"l-01957d72123243f5","kind":"oeis_match","node_a_id":"n-bb1f3df8492d87c4","node_b_id":null,"fingerprint":null,"oeis_id":"A168937","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168937","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-13595627f9023fab","kind":"oeis_match","node_a_id":"n-bb1f3df8492d87c4","node_b_id":null,"fingerprint":null,"oeis_id":"A168841","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168841","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-17f23f1d0ede2594","kind":"oeis_match","node_a_id":"n-bb1f3df8492d87c4","node_b_id":null,"fingerprint":null,"oeis_id":"A168697","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168697","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8b13411c6ecbbca2","kind":"oeis_match","node_a_id":"n-bb1f3df8492d87c4","node_b_id":null,"fingerprint":null,"oeis_id":"A167931","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A167931","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-91aacf719138a652","kind":"oeis_match","node_a_id":"n-bb1f3df8492d87c4","node_b_id":null,"fingerprint":null,"oeis_id":"A168985","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168985","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-998773b6ddf4aa27","kind":"oeis_match","node_a_id":"n-bb1f3df8492d87c4","node_b_id":null,"fingerprint":null,"oeis_id":"A168889","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168889","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-acfff26553d31c04","kind":"oeis_match","node_a_id":"n-bb1f3df8492d87c4","node_b_id":null,"fingerprint":null,"oeis_id":"A168745","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168745","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-cfbfda91292d5d1d","kind":"oeis_match","node_a_id":"n-bb1f3df8492d87c4","node_b_id":null,"fingerprint":null,"oeis_id":"A168793","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168793","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A167931","relation":"match_row_sums"},{"oeis_id":"A168697","relation":"match_row_sums"},{"oeis_id":"A168745","relation":"match_row_sums"},{"oeis_id":"A168793","relation":"match_row_sums"},{"oeis_id":"A168841","relation":"match_row_sums"},{"oeis_id":"A168889","relation":"match_row_sums"},{"oeis_id":"A168937","relation":"match_row_sums"},{"oeis_id":"A168985","relation":"match_row_sums"}],"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:          1872,           3192,            1813,             343;\n  1:         22464,          51408,           44100,           16807,            2401;\n  2:        269568,         774144,          889056,          510384,          146461,           16807;\n  3:       3234816,       11176704,        16087680,        12348000,         5330220,         1226911,          117649;\n  4:      38817792,      156764160,       271289088,       260789760,       150398640,        52034472,        10000165,          823543;\n  5:     465813504,     2152894464,      4352818176,      5028500736,      3630312000,      1677204144,       484243284,        79883671,        5764801;\n  6:    5589762048,    29095428096,     67304079360,     90811736064,     78763249152,     45538633728,     17551348416,      4348307040,      628363309,       40353607;\n  7:   67077144576,   388273471488,   1011316948992,   1560869388288,   1580841142272,   1097806348800,    529386617088,    175039123392,    37978508988,     4882786447,     282475249;\n  8:  804925734912,  5128821669888,  14853717688320,  25809651302400,  29896179425280,  24239564181504,  14037283846656,   5806175800320,  1681015971600,   324443000280,   37569208117,   1977326743;\n  9: 9659108818944, 67180340183040, 214146363949056, 413691839447040, 539421712220160, 500148026155008, 338124355430400, 167935096530432, 60815422261440, 15660427804560, 2721931499364, 286712377735, 13841287201;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 3192*x + 1813*x^2 + 343*x^3.","mathematica":"P[0, x_] := 1872 + 3192*x + 1813*x^2 + 343*x^3;\nP[n_, x_] := P[n, x] = Expand[(12 + 7*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1872 + 3192*x + 1813*x^2 + 343*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 7*x)*v[i - 1]); 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-5cf6acfdac607d6d","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-946cc0d28b50f639"],"text":"A038333 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A038333 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}