{"id":"n-a4f5036208268596","revision":0,"first_index":0,"fingerprint":"f6-72578208ad49d2f0","name":"theta_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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":[["169","624","432"],["2197","12168","16848","6912"],["28561","210912","438048","359424","103680"],["371293","3427320","9491040","11681280","6739200","1492992"],["4826809","53466192","185075280","303713280","262828800","116453376","20901888"],["62748517","810903912","3368370096","6909477120","7972473600","5298628608","1902071808","286654464"],["815730721","12047715264","58385081664","143717124096","207284313600","183685791744","98907734016","29812064256","3869835264"],["10604499373","176197835736","975864936384","2802483919872","4850452938240","5372809408512","3857401626624","1744005758976","452770725888","51597803520"],["137858491849","2545079849520","15857805216240","52046129940480","105093146995200","139693044621312","125365552865280","75573582888960","29430097182720","6707714457600","681091006464"],["1792160394037","36394641848136","251962905102480","930324572686080","2146902860044800","3329350896807936","3585454811947008","2701755588280320","1402834632376320","479601583718400","97396013924352","8916100448256"],["23298085122481","516142193482656","3930621319598688","16125625926558720","41864605770873600","74196962843148288","93221825110622208","84294774354345984","54710550662676480","24939282353356800","7596889086099456","1390911669927936","115909305827328"],["302875106592253","7269002558214072","60388636637470752","272523078158842368","786124263919737600","1567410840061507584","2250641206242164736","2374302810980745216","1849216612398465024","1053684679429324800","427958085183602688","117532036108910592","19588672684818432","1497904875307008"],["3937376385699289","101766035814997008","915894322334973072","4509018202264482816","14307461603339224320","31696530321243820032","51202087442009247744","61731873085499375616","56092903909420105728","38354122331227422720","19472092875853922304","7130276857273909248","1782569214318477312","272618687305875456","19258776968232960"],["51185893014090757","1417455498851744040","13738414835024596080","73271545786797845760","253632273877377158400","618082341264254490624","1109378561243533701120","1504714406459047280640","1562588037476702945280","1246508975764891238400","759411622158302969856","347600996792103075840","115866998930701025280","26580322012322856960","3755461508805427200","246512345193381888"],["665416609183179841","19655382917410850688","204113591834651141760","1172344732588765532160","4396292747207870745600","11687375180269539459072","23075074073865500983296","34775621838164648263680","40627288974394276577280","37039123851299625369600","26326269568154502955008","14460201466551487954944","6025083944396453314560","1842902326187718082560","390567996915764428800","51274567800223432704","3143032401215619072"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1225","38125","1140625","33203125","947265625","26611328125","738525390625","20294189453125","553131103515625","14972686767578125","402927398681640625","10788440704345703125","287592411041259765625","7636845111846923828125","202096998691558837890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["169","624","432"]],"recurrence":{"denominator":{"coeffs":[["13","24"],["0","12"]]},"terms":[{"coeff":{"coeffs":[["169","624","432"],["0","156","144"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194013,"position":0,"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-166641f50a97e101"]},{"id":194014,"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-166641f50a97e101"]},{"id":194015,"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-166641f50a97e101"]},{"id":194016,"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-166641f50a97e101"]}],"region_exact":{},"incoming_edges":[{"id":"e-166641f50a97e101","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-a4f5036208268596","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"660d0063a120c70f06f51a850a3b4d4e18b214652c377d8e4ecf666805977942","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b8a3f88eab200094","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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:           169,            624,             432;\n  1:          2197,          12168,           16848,            6912;\n  2:         28561,         210912,          438048,          359424,           103680;\n  3:        371293,        3427320,         9491040,        11681280,          6739200,          1492992;\n  4:       4826809,       53466192,       185075280,       303713280,        262828800,        116453376,         20901888;\n  5:      62748517,      810903912,      3368370096,      6909477120,       7972473600,       5298628608,       1902071808,        286654464;\n  6:     815730721,    12047715264,     58385081664,    143717124096,     207284313600,     183685791744,      98907734016,      29812064256,       3869835264;\n  7:   10604499373,   176197835736,    975864936384,   2802483919872,    4850452938240,    5372809408512,    3857401626624,    1744005758976,     452770725888,     51597803520;\n  8:  137858491849,  2545079849520,  15857805216240,  52046129940480,  105093146995200,  139693044621312,  125365552865280,   75573582888960,   29430097182720,   6707714457600,   681091006464;\n  9: 1792160394037, 36394641848136, 251962905102480, 930324572686080, 2146902860044800, 3329350896807936, 3585454811947008, 2701755588280320, 1402834632376320, 479601583718400, 97396013924352, 8916100448256;\n","recurrence_sentence":"The row polynomials satisfy (13+(12*n+24)*x)*P(n,x) = (169+(156*n+624)*x+(144*n+432)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 624*x + 432*x^2.","mathematica":"P[0, x_] := 169 + 624*x + 432*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((169 + (156*n + 624)*x + (144*n + 432)*x^2)*P[n - 1, x])/(13 + (12*n + 24)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 169 + 624*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((169 + (156*n + 624)*x + (144*n + 432)*x^2)*v[i - 1])/(13 + (12*n + 24)*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-4397e742863fdbe2","steps":["basis A013611","\u03b8+1"],"edges":["e-7b93be005ec5b750","e-166641f50a97e101"],"text":"A038305 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038305 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}