{"id":"n-9e175ec83e8af6f6","revision":0,"first_index":0,"fingerprint":"f6-9ec676f9f53aaaaa","name":"basis_map_A013610 of Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.","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":[["2"],["1"],["3","6"],["4","9"],["7","24","18"],["11","45","45"],["18","90","135","54"],["29","168","315","189"],["47","312","720","648","162"],["76","567","1539","1782","729"],["123","1020","3195","4590","2835","486"],["199","1815","6435","10989","8910","2673"],["322","3204","12690","25164","25515","11664","1458"],["521","5616","24570","55458","67392","41067","9477"],["843","9786","46872","118692","168399","129276","45927","4374"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["2","1","9","13","49","101","297","701","1889","4693","12249","31021","80017","204101","524169"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201350,"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-465ea0a86c50c551"]},{"id":201351,"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-465ea0a86c50c551"]},{"id":201352,"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-465ea0a86c50c551"]},{"id":201353,"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-465ea0a86c50c551"]}],"region_exact":{},"incoming_edges":[{"id":"e-465ea0a86c50c551","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-9e175ec83e8af6f6","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":"cdf66d4a145ae5aae25fa313254a0320a5ba5671ae3ab466fc9e93c180364b61","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-fe6a74fcc4c3def7","role":"sequence","name":"Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.","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:  2;\n  1:  1;\n  2:  3,   6;\n  3:  4,   9;\n  4:  7,  24,   18;\n  5: 11,  45,   45;\n  6: 18,  90,  135,   54;\n  7: 29, 168,  315,  189;\n  8: 47, 312,  720,  648, 162;\n  9: 76, 567, 1539, 1782, 729;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 1.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*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] = 2; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*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-fe6a74fcc4c3def7","steps":["basis A013610"],"edges":["e-465ea0a86c50c551"],"text":"A034807 \u2192 basis A013610"},"headline":"A034807 \u2192 basis A013610","chain":null}