{"id":"n-d785ee27c0a2aae2","revision":1,"first_index":0,"fingerprint":"f6-10e1deec62923f54","name":"obreschkoff_combination of Triangle read by rows: T(n,k) 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 = 8.","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","19","1"],["2","197","180","1"],["2","1815","6265","1637","1"],["2","16393","150448","145998","14758","1"],["2","147611","3085497","7362462","2955892","132855","1"],["2","1328589","58387444","284728055","280451090","55579163","1195736","1"],["2","11957407","1056624577","9488557499","18593166985","9266493853","1002374003","10761673","1"],["2","107616785","18634624128","288544971276","1000385288324","991280678838","280335102000","17644207532","96855114","1"],["2","968551203","323697607825","8267788431052","47077729726864","81511779397402","46374994019302","8006087953180","306161017078","871696091","1"],["2","8716960981","5572788729940","227593987718045","2022594350166620","5640935154859466","5606501105188928","1984487144578370","219911597372690","5267596264065","7845264892","1"],["2","78452648999","95436508679721","6094214756941025","81475315121112825","346197454027970886","550480139064242442","342613547272948578","79718421964252500","5879875948298275","90177629376637","70607384109","1"],["2","706073841177","1629340171197040","160043592330039226","3131806231987732250","19474179108498571551","46662298450181473296","46458015765145201740","19213040876346735798","3058182024780420775","154259152890420672","1539240994469402","635466457070","1"],["2","6354664570795","27766707213870985","4145379725214623242","116266139760538499156","1026427207852097553641","3549549169722636322095","5303498552404667284380","3522565333065989691906","1010345973207738550093","113368001328088117607","3992749912495399610","26228172293242760","5719198113727","1"],["2","57191981137373","472696178684083668","106315550367014384243","4204488055777021469278","51476654383215164607117","248704027153132975646264","532651174489304897552139","530897225106622866589854","246207889027919083507999","50582574549767964556180","4095265434174147974913","102350567161732855618","446474361055411703","51472783023648","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","22","380","9720","327600","13684320","681670080","39429936000","2597029344000","191870078400000","15712880287104000","1412644731353856000","138314995124866560000","14650212513932835840000","1669028883596647372800000"],"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","19","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-15"],["0","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","8","-8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162881,"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-c70944d6f4a4968f"]}],"region_exact":{},"incoming_edges":[{"id":"e-c70944d6f4a4968f","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-d785ee27c0a2aae2","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":"2fa23fdd4cb374ee4f85981775b5d1c3420a18fc496392d4a9960637663b2fee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-90af30f849fc554d","role":"sequence","name":"Triangle read by rows: T(n,k) 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 = 8.","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-90af30f849fc554d","role":"sequence","name":"Triangle read by rows: T(n,k) 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 = 8.","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,        19,            1;\n  2: 2,       197,          180,             1;\n  3: 2,      1815,         6265,          1637,              1;\n  4: 2,     16393,       150448,        145998,          14758,              1;\n  5: 2,    147611,      3085497,       7362462,        2955892,         132855,              1;\n  6: 2,   1328589,     58387444,     284728055,      280451090,       55579163,        1195736,             1;\n  7: 2,  11957407,   1056624577,    9488557499,    18593166985,     9266493853,     1002374003,      10761673,            1;\n  8: 2, 107616785,  18634624128,  288544971276,  1000385288324,   991280678838,   280335102000,   17644207532,     96855114,         1;\n  9: 2, 968551203, 323697607825, 8267788431052, 47077729726864, 81511779397402, 46374994019302, 8006087953180, 306161017078, 871696091, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(8*n-15)*x)*P(n-2,x) + (8*x-8*x^2)*d/dx P(n-2,x) + (8*n+1)*x*P(n-1,x) + (8*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 2 + 19*x + x^2.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 2 + 19*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (8*n - 15)*x)*P[n - 2, x] + (8*x - 8*x^2)*D[P[n - 2, x], x] + (8*n + 1)*x*P[n - 1, x] + (8*x - 8*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 + 19*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (8*n - 15)*x)*v[i - 2] + (8*x - 8*x^2)*deriv(v[i - 2], x) + (8*n + 1)*x*v[i - 1] + (8*x - 8*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-90af30f849fc554d","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-c70944d6f4a4968f"],"text":"A167884 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A167884 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}