{"id":"n-af589024565f1365","revision":1,"first_index":1,"fingerprint":"f6-bc743ec38800df61","name":"veronese_section_2_0 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","1"],["3","5"],["5","16","1"],["8","43","9"],["13","104","46","1"],["21","235","179","13"],["34","506","591","92","1"],["55","1051","1746","475","17"],["89","2123","4759","2010","154","1"],["144","4194","12201","7405","995","21"],["233","8136","29809","24620","5161","232","1"],["377","15546","70046","75615","22892","1803","25"],["610","29327","159387","218000","90241","11100","326","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","3","8","22","60","164","448","1224","3344","9136","24960","68192","186304","508992"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_0","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"],["2","1"],["3","5"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163258,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-918cdc60ffda35ef"]},{"id":163259,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-918cdc60ffda35ef"]},{"id":163260,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-918cdc60ffda35ef"]}],"region_exact":{},"incoming_edges":[{"id":"e-918cdc60ffda35ef","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-af589024565f1365","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"ecbf1b8a9d40d1dc316741007e9805fd6ae293d4afb2d3d0d5bed08a60b12189","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-786994051dcc7211","kind":"oeis_match","node_a_id":"n-af589024565f1365","node_b_id":null,"fingerprint":null,"oeis_id":"A155020","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A155020","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f0599e3cd5e34afe","kind":"oeis_match","node_a_id":"n-af589024565f1365","node_b_id":null,"fingerprint":null,"oeis_id":"A028859","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A028859","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A028859","relation":"match_row_sums"},{"oeis_id":"A155020","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-af589024565f1365","oeis_id":"A028859","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":31,"oeis_terms_available":31,"term_count":200,"raw_size":9682},{"node_id":"n-af589024565f1365","oeis_id":"A155020","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":32,"oeis_terms_available":32,"term_count":200,"raw_size":9596}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  1;\n   3:  2,    1;\n   4:  3,    5;\n   5:  5,   16,    1;\n   6:  8,   43,    9;\n   7: 13,  104,   46,   1;\n   8: 21,  235,  179,  13;\n   9: 34,  506,  591,  92,  1;\n  10: 55, 1051, 1746, 475, 17;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*P(n-1,x) + (1+x)*P(n-2,x) - (2-2*x)*P(n-3,x) - (1-x)*P(n-4,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + x, P(4,x) = 3 + 5*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 2 + x;\nP[4, x_] := 3 + 5*x;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] + (1 + x)*P[n - 2, x] - (2 - 2*x)*P[n - 3, x] - (1 - x)*P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 2 + x; v[4] = 3 + 5*x; for(n = 5, N, my(i = n); v[i] = 2*v[i - 1] + (1 + x)*v[i - 2] - (2 - 2*x)*v[i - 3] - (1 - x)*v[i - 4]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["Veronese 2"],"edges":["e-918cdc60ffda35ef"],"text":"A063967 \u2192 Veronese 2"},"headline":"A063967 \u2192 Veronese 2","chain":null}
