{"id":"n-b9e4c12b0bb6a69d","revision":0,"first_index":0,"fingerprint":"f6-7981e00c6bfc55e2","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["3","2"],["5","6"],["8","14","4"],["13","30","16"],["21","60","48","8"],["34","116","124","40"],["55","218","292","144","16"],["89","402","648","432","96"],["144","730","1376","1160","400","32"],["233","1310","2828","2888","1360","224"],["377","2328","5664","6800","4080","1056","64"],["610","4104","11112","15344","11216","4000","512"],["987","7186","21432","33472","28896","13216","2688","128"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","5","11","26","59","137","314","725","1667","3842","8843","20369","46898","108005"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198553,"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-6bff37bed6c42dd9"]},{"id":198554,"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-6bff37bed6c42dd9"]},{"id":198555,"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-6bff37bed6c42dd9"]},{"id":198556,"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-6bff37bed6c42dd9"]}],"region_exact":{},"incoming_edges":[{"id":"e-6bff37bed6c42dd9","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-b9e4c12b0bb6a69d","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":"371d042e1c3097e384686330a29c082acf4542c0eda75e18a8926f7bdfa35ef5","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-a5d9cd4cf05da467","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.","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:  2;\n  2:  3,   2;\n  3:  5,   6;\n  4:  8,  14,   4;\n  5: 13,  30,  16;\n  6: 21,  60,  48,   8;\n  7: 34, 116, 124,  40;\n  8: 55, 218, 292, 144, 16;\n  9: 89, 402, 648, 432, 96;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, 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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + 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-a5d9cd4cf05da467","steps":["basis A013609"],"edges":["e-6bff37bed6c42dd9"],"text":"A129710 \u2192 basis A013609"},"headline":"A129710 \u2192 basis A013609","chain":null}
