{"id":"n-e29f027434b1791d","revision":1,"first_index":0,"fingerprint":"f6-0349e4f6afd59844","name":"obreschkoff_combination of Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 9.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":true,"rows":[["-9","1"],["100","-20","1"],["-1210","341","-32","1"],["15840","-5664","758","-45","1"],["-223080","95498","-16309","1389","-59","1"],["3363360","-1661576","340924","-37190","2275","-74","1"],["-54054000","30050100","-7133460","937399","-73650","3460","-90","1"],["922521600","-566668800","151676944","-23107868","2191784","-132545","4991","-107","1"],["-16663046400","11154373200","-3304345276","568594592","-62635965","4581129","-222474","6918","-125","1"],["317578060800","-229194619200","74096101872","-14131726416","1760945696","-149813496","8813217","-354024","9294","-144","1"],["-6369204441600","4913223300000","-1714580170344","357323308676","-49415544070","4761932825","-326305392","15900723","-540030","12175","-164","1"],["134088514560000","-109787841216000","40996966492800","-9233094057440","1396875583576","-149570664100","11623386830","-660581625","27250773","-795850","15620","-185","1"],["-2956651746048000","2554575192086400","-1013513243843520","244497083470056","-40015598289612","4692346657306","-405620520525","26172444353","-1260648081","44771793","-1139655","19691","-207","1"],["68143783099392000","-61826823176025600","25908168906766080","-6646551305425824","1166303961108208","-148094058535280","14033577362376","-1008263584042","55195150439","-2291207490","70999643","-1592734","24453","-230","1"],["-1638548239071744000","1554657056528083200","-684679179288533760","185679502462226928","-34679310087289504","4725408199739864","-485317494485824","38258752667135","-2334235642882","110229710162","-3996354362","109245136","-2179814","29974","-254","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["-8","81","-900","10890","-142560","2007720","-30270240","486486000","-8302694400","149967417600","-2858202547200","57322839974400","-1206796631040000","26609865714432000","-613294047894528000"],"nonn":false,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["-9","1"]],"recurrence":{"denominator":{"coeffs":[["8","-1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["-81","18","-1"],["-18","2","0"],["-1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162844,"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-9a862b15d72a4b43"]}],"region_exact":{},"incoming_edges":[{"id":"e-9a862b15d72a4b43","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-e29f027434b1791d","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":"8f3dc56783bb64b7bbc1456e9c087f3991bcbab56ddf345ca4640a78d313be11","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-7048485c2110aad8","role":"sequence","name":"Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 9.","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-7048485c2110aad8","role":"sequence","name":"Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 9.","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:           -9,             1;\n  1:          100,           -20,           1;\n  2:        -1210,           341,         -32,            1;\n  3:        15840,         -5664,         758,          -45,          1;\n  4:      -223080,         95498,      -16309,         1389,        -59,          1;\n  5:      3363360,      -1661576,      340924,       -37190,       2275,        -74,       1;\n  6:    -54054000,      30050100,    -7133460,       937399,     -73650,       3460,     -90,       1;\n  7:    922521600,    -566668800,   151676944,    -23107868,    2191784,    -132545,    4991,    -107,    1;\n  8: -16663046400,   11154373200, -3304345276,    568594592,  -62635965,    4581129, -222474,    6918, -125,    1;\n  9: 317578060800, -229194619200, 74096101872, -14131726416, 1760945696, -149813496, 8813217, -354024, 9294, -144, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+8-x)*P(n,x) = -(n^2+18*n+81-(2*n+18)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -9 + x.","mathematica":"P[0, x_] := -9 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 18*n + 81 - (2*n + 18)*x + x^2)*P[n - 1, x])/(n + 8 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -9 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 18*n + 81 - (2*n + 18)*x + x^2)*v[i - 1])/(n + 8 - 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-4f6e747b794bcc2e","steps":["x \u21a6 \u22122x+19","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-85dd368cbda0ff5a","e-9a862b15d72a4b43"],"text":"A028338 \u2192 x \u21a6 \u22122x+19 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+19 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}