{"id":"n-80d93a39aa9a55a2","revision":0,"first_index":0,"fingerprint":"f6-eb297539291db79c","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.","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"],["3","8"],["9","44","48"],["27","188","400","256"],["81","728","2288","2944","1280"],["243","2672","11088","21504","19200","6144"],["729","9476","48832","126784","173056","115712","28672"],["2187","32804","201920","657472","1213440","1254400","659456","131072"],["6561","111536","798240","3127296","7291648","10272768","8429568","3604480","589824"],["19683","373976","3050848","13974784","39424256","70580224","79454208","53477376","19070976","2621440"],["59049","1240028","11357072","59560576","197320960","429360128","617230336","573210624","324206592","98304000","11534336"],["177147","4074380","41393328","244580736","930984704","2387579904","4187688960","4987518976","3912302592","1894514688","495976448","50331648"],["531441","13286024","148272016","974662528","4192181760","12389296128","25690247168","37455429632","37822529536","25519718400","10738466816","2457862144","218103808"],["1594323","43046720","523469360","3789174016","18175811072","60846790656","145765576704","251927592960","312343396352","272227631104","160292667392","59324235776","11995709440","939524096"],["4782969","138706100","1825499808","14428622784","76371788544","285724981248","777019428864","1554956992512","2290817630208","2457941311488","1875379224576","975087599616","320612597760","57780731904","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","101","871","7321","60851","503261","4152751","34231921","282047771","2323383701","19137123511","157620587401","1298196219971","10692116430221"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","-12","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204760,"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-eeb00a5ad79d0ea1"]},{"id":204761,"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-eeb00a5ad79d0ea1"]},{"id":204762,"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-eeb00a5ad79d0ea1"]},{"id":204763,"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-eeb00a5ad79d0ea1"]}],"region_exact":{},"incoming_edges":[{"id":"e-eeb00a5ad79d0ea1","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-80d93a39aa9a55a2","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":"275a9af80248f0f05171fd859277d6bb30f98e4ed7aaf7a95dfe199c6705c9b6","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-b1a7dcc8610d7e32","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.","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:     3,      8;\n  2:     9,     44,      48;\n  3:    27,    188,     400,      256;\n  4:    81,    728,    2288,     2944,     1280;\n  5:   243,   2672,   11088,    21504,    19200,     6144;\n  6:   729,   9476,   48832,   126784,   173056,   115712,    28672;\n  7:  2187,  32804,  201920,   657472,  1213440,  1254400,   659456,   131072;\n  8:  6561, 111536,  798240,  3127296,  7291648, 10272768,  8429568,  3604480,   589824;\n  9: 19683, 373976, 3050848, 13974784, 39424256, 70580224, 79454208, 53477376, 19070976, 2621440;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3+12*x+16*x^2)*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(3 + 12*x + 16*x^2)*P[n - 2, x] + (4 + 8*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] = 1; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 12*x + 16*x^2)*v[i - 2] + (4 + 8*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-b1a7dcc8610d7e32","steps":["basis A013611"],"edges":["e-eeb00a5ad79d0ea1"],"text":"A330381 \u2192 basis A013611"},"headline":"A330381 \u2192 basis A013611","chain":null}