{"id":"n-34e4ae47eb2b0153","revision":0,"first_index":0,"fingerprint":"f6-94dfba4114b12650","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1681","1968","576"],["68921","121032","70848","13824"],["2825761","6616416","5809536","2267136","331776"],["115856201","339091320","396984960","232381440","68014080","7962624"],["4750104241","16683292944","24414575040","19055278080","8365731840","1958805504","191102976"],["194754273881","798017512488","1401396607296","1367216202240","800321679360","281088589824","54846554112","4586471424"],["7984925229121","37392820585152","76609681198848","89689382866944","65626377707520","30732352487424","8994834874368","1504362627072","110075314176"],["327381934393961","1724743849490136","4038424623196416","5515897046317056","4843226674814976","2835059516964864","1106364689547264","277554904694784","40617790930944","2641807540224"],["13422659310152401","78571664254550640","206969261938816320","323073969855713280","330953822779023360","232474880391118848","113402380678594560","37932503641620480","8326647140843520","1083141091491840","63403380965376"],["550329031716248441","3543582057880233864","10371459681600684480","18213295050615836160","21322882010477076480","17474361842732433408","10228894737209229312","4276889785592709120","1251772620173475840","244248316131409920","28594924815384576","1521681143169024"],["22563490300366186081","158494761134279551008","510275816334753676416","995660129433665710080","1311357243644340203520","1228198003803479605248","838769368451156803584","420845954902322577408","153968032281337528320","40056723845551226880","7034351504584605696","748667122439159808","36520347436056576"],["925103102315013629321","7039808973714250057272","24725182736947609957248","53068684898814382347264","77661490095825925386240","81828692003406828699648","63866296197780939472896","37385148993822988959744","16412992241190580518912","5337558452419700981760","1249769783981198278656","199519788130036088832","19465345183418155008","876488338465357824"],["37929227194915558802161","310834642377844579451856","1182687907583994009621696","2769220466538132315211776","4457769531500408117170176","5218852134439502185955328","4582406752190782407180288","3065582217493485094699008","1570176257740565536309248","612751710337781672706048","179341964001301952987136","38174786128880238329856","5586554067641010487296","503104306279115390976","21035720123168587776"],["1555098314991537910888601","13654521790169601168777960","55950235628012024301334080","141922548910079281154603520","249229841988431908369059840","320959406268029384436252672","313131128066370131157319680","235666632969811666654986240","137951199787206829261455360","62807050309622621452369920","22059061572160140217417728","5869373367315336643215360","1145243583866407149895680","154704574180827982725120","12936967875748681482240","504857282956046106624"],["63759030914653054346432641","597157752956750557781222784","2621668183712563424405368320","7161630160385539110570762240","13624564695367610990841937920","19140851864711570562743795712","20541402001153880603920171008","17177479025355161480630108160","11311998382550959999439339520","5885917857587491381822095360","2411790731889508663771004928","770061785791772167589855232","187819947754090772582891520","33828733554207718889226240","4243325463245567526174720","331186377619166245945344","12116574790945106558976"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["4225","274625","17850625","1160290625","75418890625","4902227890625","318644812890625","20711912837890625","1346274334462890625","87507831740087890625","5688009063105712890625","369720589101871337890625","24031838291621636962890625","1562069488955406402587890625","101534516782101416168212890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1681","1968","576"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["41","24"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194835,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-2bea39d913239ab3"]},{"id":194836,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-2bea39d913239ab3","e-9ed1a0bc990308de"]},{"id":194837,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-2bea39d913239ab3","e-9ed1a0bc990308de"]},{"id":194838,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-2bea39d913239ab3"]},{"id":194839,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-9ed1a0bc990308de"]}],"region_exact":{},"incoming_edges":[{"id":"e-9ed1a0bc990308de","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-34e4ae47eb2b0153","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e5f913a161628a519995d556c14f2e74509d36f89b73bdaa84a9086007c7db31","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2117bc35fff39734","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-2bea39d913239ab3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-34e4ae47eb2b0153","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"f027a9a49011438064a46b4d3f519398f933b44713cbad162883f7dda980e219","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2117bc35fff39734","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:               1681,                1968,                  576;\n  1:              68921,              121032,                70848,                13824;\n  2:            2825761,             6616416,              5809536,              2267136,               331776;\n  3:          115856201,           339091320,            396984960,            232381440,             68014080,              7962624;\n  4:         4750104241,         16683292944,          24414575040,          19055278080,           8365731840,           1958805504,            191102976;\n  5:       194754273881,        798017512488,        1401396607296,        1367216202240,         800321679360,         281088589824,          54846554112,          4586471424;\n  6:      7984925229121,      37392820585152,       76609681198848,       89689382866944,       65626377707520,       30732352487424,        8994834874368,       1504362627072,        110075314176;\n  7:    327381934393961,    1724743849490136,     4038424623196416,     5515897046317056,     4843226674814976,     2835059516964864,     1106364689547264,     277554904694784,      40617790930944,      2641807540224;\n  8:  13422659310152401,   78571664254550640,   206969261938816320,   323073969855713280,   330953822779023360,   232474880391118848,   113402380678594560,   37932503641620480,    8326647140843520,   1083141091491840,    63403380965376;\n  9: 550329031716248441, 3543582057880233864, 10371459681600684480, 18213295050615836160, 21322882010477076480, 17474361842732433408, 10228894737209229312, 4276889785592709120, 1251772620173475840, 244248316131409920, 28594924815384576, 1521681143169024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (41+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 1681 + 1968*x + 576*x^2.","mathematica":"P[0, x_] := 1681 + 1968*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[(41 + 24*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1681 + 1968*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (41 + 24*x)*v[i - 1]); 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-1f2dec04776afb2b","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-5fcdd9bd1fb647dd","e-2bea39d913239ab3"],"text":"A038320 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A038320 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}
