{"id":"n-40814067965b299a","revision":0,"first_index":0,"fingerprint":"f6-980faab9e9709dd4","name":"basis_map_A013609 of basis_map_A013609 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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"],["8","8"],["60","112","48"],["448","1216","1024","256"],["3344","11904","14976","7680","1280"],["24960","109952","184320","144384","51200","6144"],["186304","978176","2050304","2168832","1197056","315392","28672"],["1390592","8476672","21331968","28467200","21463040","8945664","1835008","131072"],["10379520","72044544","211636224","341262336","327147520","188153856","61931520","10223616","589824"],["77473792","603236352","2026242048","3832381440","4464640000","3287744512","1506803712","404226048","55050240","2621440"],["578272256","4991430656","18870497280","40965046272","56221204480","50571902976","29863837696","11245977600","2517368832","288358400","11534336"],["4316282880","40905097216","171899092992","421283168256","665893470208","708441538560","513420558336","250548846592","79310094336","15086911488","1476395008","50331648"],["32217174016","332552962048","1537836285952","4199807057920","7515159592960","9237412446208","7946751705088","4779799478272","1971045007360","533809070080","87568678912","7415529472","218103808"],["240472260608","2685480173568","13551975530496","40811147034624","81565243146240","113791983943680","113430270836736","81260420530176","41466005028864","14700741918720","3454764318720","494659436544","36641439744","939524096"],["1794909388800","21561932840960","117911978311680","388179524321280","857148189573120","1338468700520448","1517949878796288","1263459706601472","770302405509120","339185227530240","104816109944832","21623683940352","2729787260928","178509578240","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","220","2944","39184","520960","6924736","92041216","1223368960","16260419584","216125430784","2872631787520","38181593092096","507490745122816","6745314570649600"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["8","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","-16","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["8","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200197,"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-1749b35d7a1d8bdd"]},{"id":200198,"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-1749b35d7a1d8bdd"]},{"id":200199,"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-1749b35d7a1d8bdd"]},{"id":200200,"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-1749b35d7a1d8bdd"]}],"region_exact":{},"incoming_edges":[{"id":"e-1749b35d7a1d8bdd","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-40814067965b299a","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":"a32cd79334dfb13f57f2adaffad2289df51d3d98f061ebca7a75587addf993bf","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-3d0ce3ba50a1aaca","role":"sequence","name":"basis_map_A013609 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:        8,         8;\n  2:       60,       112,         48;\n  3:      448,      1216,       1024,        256;\n  4:     3344,     11904,      14976,       7680,       1280;\n  5:    24960,    109952,     184320,     144384,      51200,       6144;\n  6:   186304,    978176,    2050304,    2168832,    1197056,     315392,      28672;\n  7:  1390592,   8476672,   21331968,   28467200,   21463040,    8945664,    1835008,    131072;\n  8: 10379520,  72044544,  211636224,  341262336,  327147520,  188153856,   61931520,  10223616,   589824;\n  9: 77473792, 603236352, 2026242048, 3832381440, 4464640000, 3287744512, 1506803712, 404226048, 55050240, 2621440;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4+16*x+16*x^2)*P(n-2,x) + (8+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(4 + 16*x + 16*x^2)*P[n - 2, x] + (8 + 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] = 8 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 16*x + 16*x^2)*v[i - 2] + (8 + 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-2245b35ec7cd4857","steps":["basis A013609","basis A013609"],"edges":["e-11b86d48c22b376e","e-1749b35d7a1d8bdd"],"text":"A091044 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A091044 \u2192 basis A013609 \u2192 basis A013609","chain":null}