{"id":"n-645e3679f0201989","revision":1,"first_index":0,"fingerprint":"f6-133d7134382ab63c","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 = 7.","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":[["-7","1"],["64","-16","1"],["-648","225","-26","1"],["7200","-3140","512","-37","1"],["-87120","45122","-9311","957","-49","1"],["1140480","-677088","166704","-21790","1595","-62","1"],["-16061760","10668216","-3020794","472759","-44170","2464","-76","1"],["242161920","-176810592","56156680","-10134964","1136954","-81193","3605","-91","1"],["-3891888000","3082525200","-1078514820","218811648","-28372043","2438569","-138950","5062","-107","1"],["66421555200","-56483020800","21476631168","-4808943344","702319148","-69911436","4804401","-225096","6882","-124","1"],["-1199739340800","1086386659200","-444200250672","108267053516","-17480242820","1963247285","-156533286","8861283","-349080","9115","-142","1"],["22865620377600","-21900839616000","9548805921984","-2506312652976","441123541056","-54854374960","4942326752","-325127913","15500793","-522390","11814","-161","1"],["-458582719795200","462028553107200","-213343772806368","59788364420520","-11346995590876","1540246721266","-153862390319","11453976545","-635499843","25957833","-758813","15035","-181","1"],["9654373048320000","-10184229315072000","4952263402713600","-1471519043729280","298534607486752","-43749066793432","4776460725036","-394730117210","24815330111","-1181145966","41904863","-1074710","18837","-202","1"],["-212878925715456000","234192493513036800","-119357383568250240","37388090347270272","-8051086739686536","1262611213719492","-148990168941614","13472311499647","-941318736358","50827030826","-2103823722","65563680","-1489306","23282","-224","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["-6","49","-448","4536","-50400","609840","-7983360","112432320","-1695133440","27243216000","-464950886400","8398175385600","-160059342643200","3210079038566400","-67580611338240000"],"nonn":false,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["-7","1"]],"recurrence":{"denominator":{"coeffs":[["6","-1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["-49","14","-1"],["-14","2","0"],["-1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162829,"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-646058a1a3e3d26a"]}],"region_exact":{},"incoming_edges":[{"id":"e-646058a1a3e3d26a","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-645e3679f0201989","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":"0eae55dbff9feeea5b979e423e67fd6a49892489e2dbe1c2bf6d92897d6a57a5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-0daa6c2f3598d022","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 = 7.","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-0daa6c2f3598d022","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 = 7.","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:          -7,            1;\n  1:          64,          -16,           1;\n  2:        -648,          225,         -26,           1;\n  3:        7200,        -3140,         512,         -37,         1;\n  4:      -87120,        45122,       -9311,         957,       -49,         1;\n  5:     1140480,      -677088,      166704,      -21790,      1595,       -62,       1;\n  6:   -16061760,     10668216,    -3020794,      472759,    -44170,      2464,     -76,       1;\n  7:   242161920,   -176810592,    56156680,   -10134964,   1136954,    -81193,    3605,     -91,    1;\n  8: -3891888000,   3082525200, -1078514820,   218811648, -28372043,   2438569, -138950,    5062, -107,    1;\n  9: 66421555200, -56483020800, 21476631168, -4808943344, 702319148, -69911436, 4804401, -225096, 6882, -124, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+6-x)*P(n,x) = -(n^2+14*n+49-(2*n+14)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -7 + x.","mathematica":"P[0, x_] := -7 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 14*n + 49 - (2*n + 14)*x + x^2)*P[n - 1, x])/(n + 6 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -7 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 14*n + 49 - (2*n + 14)*x + x^2)*v[i - 1])/(n + 6 - 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+15","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a85f747dc534a48f","e-646058a1a3e3d26a"],"text":"A028338 \u2192 x \u21a6 \u22122x+15 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+15 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}