{"id":"n-88f2be16f47e95a3","revision":1,"first_index":1,"fingerprint":"f6-d59460fefeb34cde","name":"obreschkoff_combination of Triangle read by rows: 3-Stirling numbers of the second kind.","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":[["4","1"],["12","8","1"],["36","44","13","1"],["108","212","109","19","1"],["324","956","757","223","26","1"],["972","4148","4741","2095","405","34","1"],["2916","17564","27853","17311","4930","677","43","1"],["8748","73172","156829","131719","51821","10346","1064","53","1"],["26244","301436","857317","947143","494466","134589","19922","1594","64","1"],["78732","1231988","4588021","6540175","4408405","1571178","313887","35862","2298","76","1"],["236196","5006684","24172093","43829071","37399010","16977829","4396161","672507","61140","3210","89","1"],["708588","20262932","125867149","287146519","305622141","173221642","56543278","11121231","1345047","99660","4367","103","1"],["2125764","81760316","649598677","1848746263","2426501506","1691395277","682111144","167755588","25916748","2540967","156431","5809","118","1"],["6377292","329167028","3329753701","11742076255","18834256805","15957663722","7830395573","2359667024","452839816","56408352","4574570","237757","7579","134","1"],["19131876","1323045404","16977935533","73782211231","143581873890","146495566581","86431223879","31427065813","7340905000","1129740040","115877762","7903168","351442","9723","151","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","21","94","449","2287","12396","71295","433753","2782776","18770623","132753991","981942658","7578614609","60903425609","508632841494"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["4","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162747,"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-0e86cfd28f077eaa"]}],"region_exact":{},"incoming_edges":[{"id":"e-0e86cfd28f077eaa","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-88f2be16f47e95a3","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":"84e4abdebe914b381c66749f3eb3cd79bf86796653c93cc748533b46a2c49629","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-a2b8ff03da0c14a3","role":"sequence","name":"Triangle read by rows: 3-Stirling numbers of the second kind.","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-a2b8ff03da0c14a3","role":"sequence","name":"Triangle read by rows: 3-Stirling numbers of the second kind.","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:     4,       1;\n   2:    12,       8,       1;\n   3:    36,      44,      13,       1;\n   4:   108,     212,     109,      19,       1;\n   5:   324,     956,     757,     223,      26,       1;\n   6:   972,    4148,    4741,    2095,     405,      34,      1;\n   7:  2916,   17564,   27853,   17311,    4930,     677,     43,     1;\n   8:  8748,   73172,  156829,  131719,   51821,   10346,   1064,    53,    1;\n   9: 26244,  301436,  857317,  947143,  494466,  134589,  19922,  1594,   64,  1;\n  10: 78732, 1231988, 4588021, 6540175, 4408405, 1571178, 313887, 35862, 2298, 76, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + x.","mathematica":"P[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + 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] = 4 + x; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + 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-a2b8ff03da0c14a3","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0e86cfd28f077eaa"],"text":"A143495 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A143495 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}