{"id":"n-7b9e0785bd039123","revision":1,"first_index":1,"fingerprint":"f6-58d17f888e14c685","name":"obreschkoff_combination of Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2","1"],["2","8","1"],["2","50","19","1"],["2","302","259","35","1"],["2","1814","3151","819","56","1"],["2","10886","36475","16255","1995","82","1"],["2","65318","412111","296555","58150","4127","113","1"],["2","391910","4598539","5156991","1517705","165452","7630","149","1"],["2","2351462","50975839","87110395","37028796","5819457","401982","12994","190","1"],["2","14108774","563085691","1444742159","864715111","188334678","18280899","869766","20784","236","1"],["2","84652646","6208051375","23678960235","19603759490","5761416739","755042547","49592475","1721910","31640","287","1"],["2","507915878","68373217771","385071415135","435357909525","169400594704","29167735696","2540371647","120190785","3177350","46277","343","1"],["2","3047495270","752613311359","6229515859931","9527587515160","4839773371829","1073600401280","120621114988","7468193832","266348885","5537477","65485","404","1"],["2","18284971622","8281793920219","100424867070255","206308853678291","135361695182714","38121385811509","5415960540848","426817062100","19720242542","548760212","9204637","90129","470","1"],["2","109709829734","91118018094031","1615079667044299","4432910794314366","3725712928428855","1317124655339493","233095965282037","22915460086948","1333948219032","47707013354","1064219884","14702506","121149","541","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","11","72","599","5843","65696","836377","11838379","183701118","3094158101","56143229347","1090542575114","22554499215903","494379936535551","11439449932696232"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162783,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-61c0915f6ff6bb4a"]}],"region_exact":{},"incoming_edges":[{"id":"e-61c0915f6ff6bb4a","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-7b9e0785bd039123","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":"b22e262f010597573328d9dc1674c0bce907b995733c63707c9d552fb8e42b90","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-7466465df6cec0d5","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-7466465df6cec0d5","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 2,        1;\n   2: 2,        8,         1;\n   3: 2,       50,        19,          1;\n   4: 2,      302,       259,         35,         1;\n   5: 2,     1814,      3151,        819,        56,         1;\n   6: 2,    10886,     36475,      16255,      1995,        82,        1;\n   7: 2,    65318,    412111,     296555,     58150,      4127,      113,      1;\n   8: 2,   391910,   4598539,    5156991,   1517705,    165452,     7630,    149,     1;\n   9: 2,  2351462,  50975839,   87110395,  37028796,   5819457,   401982,  12994,   190,   1;\n  10: 2, 14108774, 563085691, 1444742159, 864715111, 188334678, 18280899, 869766, 20784, 236, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 + x.","mathematica":"P[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + x; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 5*x*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7466465df6cec0d5","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-61c0915f6ff6bb4a"],"text":"A166973 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A166973 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}