{"id":"n-dd0715c4bad37c11","revision":0,"first_index":0,"fingerprint":"f6-3125eed2334c9e3a","name":"theta_plus_one_squared of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["19","32"],["361","1216","576"],["6859","34656","32832","8192"],["130321","877952","1247616","622592","102400"],["2476099","20851360","39507840","29573120","9728000","1179648"],["47045881","475411008","1125973440","1123778560","554496000","134479872","12845056"],["893871739","10538277344","29950893504","37365637120","24582656000","8942911488","1708392448","134217728"],["16983563041","228831165184","758755968768","1135915368448","934140928000","453107515392","129837826048","20401094656","1358954496"],["322687697779","4891266155808","18535324379904","32373588000768","31947619737600","19370346283008","7400756084736","1744293593088","232381218816","13421772800"],["6131066257801","103260063289280","440213954022720","878711674306560","1011674625024000","736073158754304","351535914024960","110471927562240","22076215787520","2550136832000","129922760704"],["116490258898219","2158135322745952","10222746265638720","22956342491258880","30205713804288000","25639881696608256","14694201206243328","5772158215127040","1537976366530560","266489298944000","27153856987136","1236950581248"],["2213314919066161","44732259416916096","233078614856562816","581560676445224960","860862843422208000","835127575260954624","558379645837246464","263210414609793024","87664652892241920","20253186719744000","3095539696533504","282024732524544","11613591568384"],["42052983462257059","920739006331522976","5233674351779183232","14364548708197056512","23625902480587264000","25784563886181974016","19702824645971410944","10835495401436479488","4330633852876750848","1250634279944192000","254866101681258496","34830054466781184","2868557117390848","107752139522048"],["799006685782884121","18839736591091162432","116013114797771894976","347360905125492457472","628449005983621222400","762077110413822787584","655118919478549413888","411748825254586220544","191991434144202620928","66533743693031014400","16948595761803689984","3088264829387931648","381518096612982784","28662069112864768","989560464998400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","51","2153","82539","2980881","103316067","3474029817","114116857371","3679332384033","116831684924307","3662828658622089","113598525733851339","3490472362400584497","106383742062173043651","3219379832485334146329"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["361","-152","0"],["0","456","0"],["0","0","64"]]},"terms":[{"coeff":{"coeffs":[["6859","8664","3648","512"],["0","8664","6080","1024"],["0","0","1216","512"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191881,"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-ecedcbcc5733b91d"]},{"id":191882,"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-ecedcbcc5733b91d"]},{"id":191883,"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-ecedcbcc5733b91d"]},{"id":191884,"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-ecedcbcc5733b91d"]}],"region_exact":{},"incoming_edges":[{"id":"e-ecedcbcc5733b91d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-dd0715c4bad37c11","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":"ee65b5782bbcd1d56e477a2d530a5e2428c5f0a9dc329e70d7871b3a93b6ef96","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-0d855b65b3c75e99","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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:            1;\n  1:           19,            32;\n  2:          361,          1216,            576;\n  3:         6859,         34656,          32832,           8192;\n  4:       130321,        877952,        1247616,         622592,         102400;\n  5:      2476099,      20851360,       39507840,       29573120,        9728000,        1179648;\n  6:     47045881,     475411008,     1125973440,     1123778560,      554496000,      134479872,      12845056;\n  7:    893871739,   10538277344,    29950893504,    37365637120,    24582656000,     8942911488,    1708392448,     134217728;\n  8:  16983563041,  228831165184,   758755968768,  1135915368448,   934140928000,   453107515392,  129837826048,   20401094656,   1358954496;\n  9: 322687697779, 4891266155808, 18535324379904, 32373588000768, 31947619737600, 19370346283008, 7400756084736, 1744293593088, 232381218816, 13421772800;\n","recurrence_sentence":"The row polynomials satisfy (361+(456*n-152)*x+64*n^2*x^2)*P(n,x) = (6859+(8664*n+8664)*x+(1216*n^2+6080*n+3648)*x^2+(512*n^2+1024*n+512)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((6859 + (8664*n + 8664)*x + (1216*n^2 + 6080*n + 3648)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*P[n - 1, x])/(361 + (456*n - 152)*x + 64*n^2*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6859 + (8664*n + 8664)*x + (1216*n^2 + 6080*n + 3648)*x^2 + (512*n^2 + 1024*n + 512)*x^3)*v[i - 1])/(361 + (456*n - 152)*x + 64*n^2*x^2)); 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-2f02b2035f8f4541","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-c22af348c5e552a2","e-ecedcbcc5733b91d"],"text":"A038322 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A038322 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}