{"id":"n-f2b1db8caed04b08","revision":0,"first_index":0,"fingerprint":"f6-65bc1c81d6788de0","name":"theta_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":[["289","1632","1728"],["4913","41616","88128","55296"],["83521","943296","2996352","3760128","1658880"],["1419857","20045040","84896640","159805440","141004800","47775744"],["24137569","408918816","2164864320","5433384960","7191244800","4873125888","1337720832"],["410338673","8110223184","51523770816","161643202560","285252710400","289950990336","159188779008","36691771392"],["6975757441","157570050432","1167872138496","4396695109632","9698592153600","13144444895232","10824836972544","4990080909312","990677827584"],["118587876497","3013527214512","25526348169984","112115725295616","296776919900160","502775017242624","552066685599744","381741189562368","151573707620352","26418075402240"],["2015993900449","56922180718560","542434898612160","2722810471464960","8408679397171200","17094350586249216","23462834137989120","21632000741867520","12883765147729920","4491072818380800","697437190619136"],["34271896307633","1064444779437072","11270591782274880","63645694770493440","224631863895859200","532773926604767232","877509996760793088","1011296034682306560","803088027541831680","419915308518604800","130420754645778432","18260173718028288"],["582622237229761","19740612273196608","229920072358407552","1442635748131184640","5728112529344409600","15526554432481787904","29835339889866964992","41260878215038107648","40957489404633415680","28554240979265126400","13302916973869400064","3725075438477770752","474764516668735488"],["9904578032905937","363556276031370864","4619303271928006272","31882250033699180544","140656985442790502400","428921066197309390848","941944302237228466176","1519775680920570298368","1810321031684796973056","1577621814104398233600","979981550408379138048","411620835951793668096","104922958183790542848","12270836738515009536"],["168377826559400929","6655876438112789664","91616181559905457728","689815955274582269952","3347636253538413957120","11342579306106626113536","28022842991557546868736","51672373151299390144512","71809400923496946597888","75094798351369355919360","58308902249298558713856","32655252985508964335616","12485832023871074598912","2920459143766572269568","315535801847528816640"],["2862423051509815793","121232035122768668880","1797086638290453209280","14658589049584873236480","77604294968390505369600","289235772305718965895168","793980551427463827947520","1647056894197668060856320","2615913890784531626065920","3191528929933197626572800","2973754014714226494406656","2081772377826196476395520","1061295722029041340907520","372358540830237964369920","80461629471119848243200","8077716527296737705984"],["48661191875666868481","2198340903559538529024","34914826115357376637440","306702786268237347717120","1759030685950184788377600","7152011824286868974862336","21596270998827016120172544","49777719469085079172546560","88941072286674075286241280","124013695563118536346828800","134810182000378267746435072","113248417353745088315916288","72168109097974811181711360","33760507701941575436206080","10942781608072299361075200","2197138895424712656027648","205981771446066811502592"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["3649","189953","9442177","454947521","21433397185","992771786369","45377745814273","2052125783884097","91994323564703041","4093911089596482305","181058251434645737089","7964912076029263686593","348763869572760138254017","15209620105161139079666561","660916783871403612127655425"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["289","1632","1728"]],"recurrence":{"denominator":{"coeffs":[["17","48"],["0","24"]]},"terms":[{"coeff":{"coeffs":[["289","1632","1728"],["0","408","576"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194831,"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-9d28750e02721f22"]},{"id":194832,"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-9d28750e02721f22"]},{"id":194833,"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-9d28750e02721f22"]},{"id":194834,"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-9d28750e02721f22"]}],"region_exact":{},"incoming_edges":[{"id":"e-9d28750e02721f22","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-f2b1db8caed04b08","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":"bbfab1d7ac3e25dd0a7ee5816758b6610a2738e3ac845a78563505505fcf8ef1","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-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:            289,             1632,              1728;\n  1:           4913,            41616,             88128,             55296;\n  2:          83521,           943296,           2996352,           3760128,            1658880;\n  3:        1419857,         20045040,          84896640,         159805440,          141004800,           47775744;\n  4:       24137569,        408918816,        2164864320,        5433384960,         7191244800,         4873125888,         1337720832;\n  5:      410338673,       8110223184,       51523770816,      161643202560,       285252710400,       289950990336,       159188779008,         36691771392;\n  6:     6975757441,     157570050432,     1167872138496,     4396695109632,      9698592153600,     13144444895232,     10824836972544,       4990080909312,       990677827584;\n  7:   118587876497,    3013527214512,    25526348169984,   112115725295616,    296776919900160,    502775017242624,    552066685599744,     381741189562368,    151573707620352,     26418075402240;\n  8:  2015993900449,   56922180718560,   542434898612160,  2722810471464960,   8408679397171200,  17094350586249216,  23462834137989120,   21632000741867520,  12883765147729920,   4491072818380800,    697437190619136;\n  9: 34271896307633, 1064444779437072, 11270591782274880, 63645694770493440, 224631863895859200, 532773926604767232, 877509996760793088, 1011296034682306560, 803088027541831680, 419915308518604800, 130420754645778432, 18260173718028288;\n","recurrence_sentence":"The row polynomials satisfy (17+(24*n+48)*x)*P(n,x) = (289+(408*n+1632)*x+(576*n+1728)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1632*x + 1728*x^2.","mathematica":"P[0, x_] := 289 + 1632*x + 1728*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((289 + (408*n + 1632)*x + (576*n + 1728)*x^2)*P[n - 1, x])/(17 + (24*n + 48)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1632*x + 1728*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (408*n + 1632)*x + (576*n + 1728)*x^2)*v[i - 1])/(17 + (24*n + 48)*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-1f2dec04776afb2b","steps":["basis A013611","\u03b8+1"],"edges":["e-5fcdd9bd1fb647dd","e-9d28750e02721f22"],"text":"A038320 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038320 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}