{"id":"n-b018e3c7840cd649","revision":1,"first_index":0,"fingerprint":"f6-f0f0549965a63c8b","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","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":[["5","4"],["20","52","16"],["80","464","432","64"],["320","3552","7072","3200","256"],["1280","25024","91264","87744","22144","1024"],["5120","167552","1024128","1776768","961408","146688","4096"],["20480","1085184","10494976","29888000","29135360","9701120","943616","16384"],["81920","6871552","101031424","443930624","704896000","425290240","92415488","5948416","65536"],["327680","42834944","930060288","6036698112","14617389056","14356423680","5712439296","844516352","36968448","262144"],["1310720","264087552","8285317120","76925114368","271164368896","404279977984","262844643328","72142735360","7483635712","227446784","1048576"],["5242880","1615458304","72018657280","933280317440","4627382558720","9970668216320","9837552607232","4441463029760","869422899200","64799109120","1389322240","4194304"],["20971520","9826967552","614465888256","10902320168960","74051588751360","222367793725440","316150854598656","217157623676928","70543406039040","10104915927040","551282237440","8442888192","16777216"],["83886080","59540619264","5168451715072","123641239158784","1126746156400640","4585440569425920","9041873302781952","8948090241908736","4442334534500352","1067105070530560","114150789120000","4626996445184","51118702592","67108864"],["335544320","359726743552","42997873672192","1369692348252160","16468262928613376","88845220553031680","235953209303433216","323206202753089536","231590559630950400","85529632174931968","15524799097241600","1260710010847232","38430417846272","308691927040","268435456"],["1342177280","2168963661824","354672587833344","14891626829840384","233003954407604224","1636957268576108544","5721100894812372992","10519305230070054912","10432052236620398592","5578355555734454272","1567895899329789952","218848847658614784","13673914692665344","316601896730624","1860607279104","1073741824"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["9","88","1040","14400","228480","4085760","81285120","1780531200","42577920000","1103619686400","30819601612800","922462558617600","29459288162304000","999830386114560000","35936760735203328000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["5","4"],["20","52","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","0"],["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162845,"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-2b0de764d6dd6b90"]}],"region_exact":{},"incoming_edges":[{"id":"e-2b0de764d6dd6b90","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-b018e3c7840cd649","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":"6f46b8e499af29750106f0f51fa1466f3729f422bbee8b9b109ba79f3dbbb6db","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c35e92273dac3e06","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","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-c35e92273dac3e06","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.","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:       5,         4;\n  1:      20,        52,         16;\n  2:      80,       464,        432,          64;\n  3:     320,      3552,       7072,        3200,          256;\n  4:    1280,     25024,      91264,       87744,        22144,         1024;\n  5:    5120,    167552,    1024128,     1776768,       961408,       146688,         4096;\n  6:   20480,   1085184,   10494976,    29888000,     29135360,      9701120,       943616,       16384;\n  7:   81920,   6871552,  101031424,   443930624,    704896000,    425290240,     92415488,     5948416,      65536;\n  8:  327680,  42834944,  930060288,  6036698112,  14617389056,  14356423680,   5712439296,   844516352,   36968448,    262144;\n  9: 1310720, 264087552, 8285317120, 76925114368, 271164368896, 404279977984, 262844643328, 72142735360, 7483635712, 227446784, 1048576;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+2*n*x)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + (3+(2*n+4)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 5 + 4*x, P(1,x) = 20 + 52*x + 16*x^2.","mathematica":"P[0, x_] := 5 + 4*x;\nP[1, x_] := 20 + 52*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(4 + 2*n*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + (3 + (2*n + 4)*x)*P[n - 1, x] + (2*x - 2*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] = 5 + 4*x; v[2] = 20 + 52*x + 16*x^2; for(n = 2, N, my(i = n + 1); v[i] = (4 + 2*n*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + (3 + (2*n + 4)*x)*v[i - 1] + (2*x - 2*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-c35e92273dac3e06","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-2b0de764d6dd6b90"],"text":"A257613 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257613 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}