{"id":"n-2400b91caa82633b","revision":0,"first_index":0,"fingerprint":"f6-80cbfc97ad0801be","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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":[["121","308","147"],["1331","5082","4851","1372"],["14641","74536","106722","60368","12005"],["161051","1024870","1956570","1660120","660275","100842"],["1771561","13528284","32283405","36522640","21789075","6655572","823543"],["19487171","173612978","497164437","703060820","559252925","256239522","63412811","6588344"],["214358881","2182563152","7291745076","12373870432","12303564350","7516359312","2790163684","579774272","51883209"],["2357947691","27009219006","103126108932","204168862128","243610574130","186029892972","92075401572","28698826464","5136437691","403536070"],["25937424601","330112676740","1417983997815","3208367833440","4466193859050","4092657645384","2532073543230","1052290303680","282504073005","44388967700","3107227739"],["285311670611","3994363388554","19064007081735","48526563480780","77201350992150","82535262515244","61276179746166","31831781686320","11394330944535","2685532545850","375974556419","23727920916"],["3138428376721","47932360662648","251644893478902","711722931051440","1273822291370475","1556379236001744","1348075954415652","840359036518848","376012921169655","118163432017400","24814320723654","3132085560912","179936733613"],["34522712143931","571193964563222","3271383615225726","10177637914035592","20239620851775325","27820278843531174","27539265925919748","20028557037032544","10753969545452133","4224342694622050","1182815954494174","223944117605208","25730952906659","1356446145698"],["379749833583241","6766451580210476","41982756395396817","142486930796498288","311690161117340005","476035882433755644","530130869073955149","440628254814715968","276018551666604747","130109754994359140","45538414248025699","11495798037067344","1981283373812743","208892706437492","10173346092735"],["4177248169415651","79747465052480610","532858061941574985","1959195298451851460","4675352416760100075","7854592060156968126","9719065933022511065","9087957755553516840","6506151574998540465","3578018262344876350","1502767670184848067","474201669029027940","108970585559700865","17233648281093090","1678602105301275","75960984159088"],["45949729863572161","935703589949105824","6698787064408371240","26524490194425065920","68571835445814801100","125673472962511490016","171055560421196194744","177720062775268773760","143135334649967890230","89961602024671176800","44081184992088876632","16691898749821783488","4794705764626838060","1011040699157461280","147716985266512200","13369133211999488","564959819683217"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["576","12636","268272","5563728","113374080","2278819008","45304282368","892616806656","17455617552384","339194386529280","6555377828081664","126094620575453184","2415463924417855488","46102044212595965952","877063280142069596160"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["121","308","147"]],"recurrence":{"denominator":{"coeffs":[["11","14"],["0","7"]]},"terms":[{"coeff":{"coeffs":[["121","308","147"],["0","77","49"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192782,"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-5727becbc977d187"]},{"id":192783,"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-5727becbc977d187"]},{"id":192784,"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-5727becbc977d187"]},{"id":192785,"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-5727becbc977d187"]}],"region_exact":{},"incoming_edges":[{"id":"e-5727becbc977d187","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-2400b91caa82633b","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":"97deb43e26dea9a6dd2ee9b5369d87da66525d8b725da9601ba3a6f2cbb9d5e2","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-7dcf11eddf2b9a8a","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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:          121,           308,            147;\n  1:         1331,          5082,           4851,           1372;\n  2:        14641,         74536,         106722,          60368,          12005;\n  3:       161051,       1024870,        1956570,        1660120,         660275,         100842;\n  4:      1771561,      13528284,       32283405,       36522640,       21789075,        6655572,         823543;\n  5:     19487171,     173612978,      497164437,      703060820,      559252925,      256239522,       63412811,        6588344;\n  6:    214358881,    2182563152,     7291745076,    12373870432,    12303564350,     7516359312,     2790163684,      579774272,       51883209;\n  7:   2357947691,   27009219006,   103126108932,   204168862128,   243610574130,   186029892972,    92075401572,    28698826464,     5136437691,     403536070;\n  8:  25937424601,  330112676740,  1417983997815,  3208367833440,  4466193859050,  4092657645384,  2532073543230,  1052290303680,   282504073005,   44388967700,   3107227739;\n  9: 285311670611, 3994363388554, 19064007081735, 48526563480780, 77201350992150, 82535262515244, 61276179746166, 31831781686320, 11394330944535, 2685532545850, 375974556419, 23727920916;\n","recurrence_sentence":"The row polynomials satisfy (11+(7*n+14)*x)*P(n,x) = (121+(77*n+308)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 308*x + 147*x^2.","mathematica":"P[0, x_] := 121 + 308*x + 147*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((121 + (77*n + 308)*x + (49*n + 147)*x^2)*P[n - 1, x])/(11 + (7*n + 14)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 121 + 308*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (77*n + 308)*x + (49*n + 147)*x^2)*v[i - 1])/(11 + (7*n + 14)*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-7dcf11eddf2b9a8a","steps":["\u03b8+1"],"edges":["e-5727becbc977d187"],"text":"A038321 \u2192 \u03b8+1"},"headline":"A038321 \u2192 \u03b8+1","chain":null}